A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calculation of the Taylor coefficients of the function with, 3, variables , x[1], x[2], x[3] 2 2 2 exp(x[1] - 8 x[1] x[2] + 21 x[1] x[3] + x[2] - 56 x[2] x[3] + x[3] ) By Shalosh B. Ekhad Theorem: Let , F(m[1], m[2], m[3]), be the coefficient of, m[1] m[2] m[3] x[1] x[2] x[3] , in the Taylor expansion (around the origing)\ of the multivariable function (that may also be viewed as a formal powe\ r series) of, x[1], x[2], x[3] 2 2 2 exp(x[1] - 8 x[1] x[2] + 21 x[1] x[3] + x[2] - 56 x[2] x[3] + x[3] ) The following PURE recurrence relations hold in each of the, 3, discrete variables, m[1], m[2], m[3] 2 F(m[1], m[2], m[3]) = -1/54 (23454508350 m[1] + 49733397520 m[1] m[2] 2 - 48151028940 m[1] m[3] + 23817582080 m[2] - 47181346576 m[2] m[3] 2 + 23243403918 m[3] - 139477559325 m[1] - 146777624016 m[2] + 144498985953 m[3] + 185419064025) F(-2 + m[1], m[2], m[3])/(m[1] 3 2 (-1 + m[1]) %1) - 1/108 (1306180007535 m[1] + 1609870124320 m[1] m[2] 2 2 - 1992165320733 m[1] m[3] - 1574217262080 m[1] m[2] 2 + 2179572327296 m[1] m[2] m[3] - 673682825067 m[1] m[3] 3 2 - 1791143936000 m[2] + 4936840473600 m[2] m[3] 2 3 2 - 4535722185120 m[2] m[3] + 1389064919193 m[3] - 13227424615938 m[1] - 13537870045344 m[1] m[2] + 15734055771000 m[1] m[3] 2 2 + 3282606074880 m[2] - 3536107041376 m[2] m[3] + 477945427386 m[3] + 42746870706267 m[1] + 28027536747904 m[2] - 30744792634899 m[3] - 42955504283952) F(m[1] - 4, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) 2 (-2 + m[1]) %1) - 1/54 (863879368170 m[1] - 2423383228400 m[1] m[2] 2 - 4466416169796 m[1] m[3] - 5036164037120 m[2] - 47181346576 m[2] m[3] 2 + 4294381794906 m[3] - 9475174874841 m[1] - 4126674053936 m[2] + 16668541495221 m[3] + 14845560932475) F(m[1] - 6, m[2], m[3])/( (-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) + 434815/9 (1736085 m[1] - 3340233 + 2412640 m[2] - 2216613 m[3]) F(m[1] - 8, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) %1 := 1736085 m[1] - 6812403 + 2412640 m[2] - 2216613 m[3] 2 F(m[1], m[2], m[3]) = -4/437 (29637661280 m[1] + 42080487670 m[1] m[2] 2 - 40770245376 m[1] m[3] + 10990022400 m[2] - 21375234230 m[2] m[3] 2 + 10392987808 m[3] - 87621119391 m[1] - 45559112240 m[2] + 44835331583 m[3] + 46153846840) F(m[1], -2 + m[2], m[3])/(m[2] 3 2 (-1 + m[2]) %1) + 4/437 (968535968000 m[1] + 488437191040 m[1] m[2] 2 2 - 981895084800 m[1] m[3] - 1635250735310 m[1] m[2] 2 + 1350617347072 m[1] m[2] m[3] + 331812821760 m[1] m[3] 3 2 2 - 582516878080 m[2] + 1132445462414 m[2] m[3] - 512193766784 m[2] m[3] 3 2 - 37376616704 m[3] + 668241914560 m[1] + 13073338095042 m[1] m[2] 2 - 6334206674432 m[1] m[3] + 5042219493104 m[2] - 7101381415170 m[2] m[3] 2 + 2064213535552 m[3] - 24716581982847 m[1] - 12298402469136 m[2] + 10351727243487 m[3] + 5878846055736) F(m[1], m[2] - 4, m[3])/( 1392 2 (-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1) + ---- (79987711840 m[1] 437 2 - 32430388250 m[1] m[2] + 43790263552 m[1] m[3] - 20708522240 m[2] 2 + 44717246042 m[2] m[3] - 23932476512 m[3] + 378920988797 m[1] + 132206558032 m[2] - 132368969117 m[3] - 9722409000) F(m[1], m[2] - 6, m[3])/((-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1) + 1084486320 ---------- 437 (179945 m[1] + 62080 m[2] - 7944 - 60809 m[3]) F(m[1], m[2] - 8, m[3]) ---------------------------------------------------------------------- (-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1 %1 := 179945 m[1] + 62080 m[2] - 132104 - 60809 m[3] 2 F(m[1], m[2], m[3]) = -1/30 (109970624442 m[1] - 187491839920 m[1] m[2] 2 + 221620100988 m[1] m[3] + 54084735488 m[2] - 113776858448 m[2] m[3] 2 + 57001465290 m[3] - 513368901669 m[1] + 250709063184 m[2] - 228952997295 m[3] + 195218597475) F(m[1], m[2], -2 + m[3])/(m[3] 3 2 (-1 + m[3]) %1) + 1/60 (57770559721917 m[1] - 63746824520736 m[1] m[2] 2 2 + 34713305811177 m[1] m[3] + 23447107869696 m[1] m[2] 2 + 95882974929280 m[1] m[2] m[3] - 124769723470545 m[1] m[3] 3 2 - 2874741194752 m[2] - 39963618358272 m[2] m[3] 2 3 2 + 82640712705440 m[2] m[3] - 39340627389525 m[3] + 13929476152578 m[1] - 435214170591200 m[1] m[2] + 996108111482904 m[1] m[3] 2 + 158194274982912 m[2] - 528746223853344 m[2] m[3] 2 + 331529212887510 m[3] - 1913569480337919 m[1] + 803565248914304 m[2] - 763561934948505 m[3] + 268347648848400) F(m[1], m[2], m[3] - 4)/( 29 2 (-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) + -- (220412596064682 m[1] 30 + 135931583942000 m[1] m[2] - 145239591219684 m[1] m[3] 2 2 - 79817152604672 m[2] + 143635770208144 m[2] m[3] - 64147918889670 m[3] + 1286950598974365 m[1] - 465981234772400 m[2] + 381685510628895 m[3] + 34488647137395) F(m[1], m[2], m[3] - 6)/((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) + 658222947/5 (3178893 m[1] - 1169248 m[2] + 962115 m[3] + 78705) F(m[1], m[2], m[3] - 8) /((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) %1 := 3178893 m[1] - 1169248 m[2] + 962115 m[3] - 1845525 and in Maple notation F(m[1],m[2],m[3]) = -1/54*(23454508350*m[1]^2+49733397520*m[1]*m[2]-48151028940 *m[1]*m[3]+23817582080*m[2]^2-47181346576*m[2]*m[3]+23243403918*m[3]^2-\ 139477559325*m[1]-146777624016*m[2]+144498985953*m[3]+185419064025)/m[1]/(-1+m[ 1])/(1736085*m[1]-6812403+2412640*m[2]-2216613*m[3])*F(-2+m[1],m[2],m[3])-1/108 *(1306180007535*m[1]^3+1609870124320*m[1]^2*m[2]-1992165320733*m[1]^2*m[3]-\ 1574217262080*m[1]*m[2]^2+2179572327296*m[1]*m[2]*m[3]-673682825067*m[1]*m[3]^2 -1791143936000*m[2]^3+4936840473600*m[2]^2*m[3]-4535722185120*m[2]*m[3]^2+ 1389064919193*m[3]^3-13227424615938*m[1]^2-13537870045344*m[1]*m[2]+ 15734055771000*m[1]*m[3]+3282606074880*m[2]^2-3536107041376*m[2]*m[3]+ 477945427386*m[3]^2+42746870706267*m[1]+28027536747904*m[2]-30744792634899*m[3] -42955504283952)/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[1])/(1736085*m[1]-6812403+ 2412640*m[2]-2216613*m[3])*F(m[1]-4,m[2],m[3])-1/54*(863879368170*m[1]^2-\ 2423383228400*m[1]*m[2]-4466416169796*m[1]*m[3]-5036164037120*m[2]^2-\ 47181346576*m[2]*m[3]+4294381794906*m[3]^2-9475174874841*m[1]-4126674053936*m[2 ]+16668541495221*m[3]+14845560932475)/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[1])/( 1736085*m[1]-6812403+2412640*m[2]-2216613*m[3])*F(m[1]-6,m[2],m[3])+434815/9*( 1736085*m[1]-3340233+2412640*m[2]-2216613*m[3])/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[ 1])/(1736085*m[1]-6812403+2412640*m[2]-2216613*m[3])*F(m[1]-8,m[2],m[3]) F(m[1],m[2],m[3]) = -4/437*(29637661280*m[1]^2+42080487670*m[1]*m[2]-\ 40770245376*m[1]*m[3]+10990022400*m[2]^2-21375234230*m[2]*m[3]+10392987808*m[3] ^2-87621119391*m[1]-45559112240*m[2]+44835331583*m[3]+46153846840)/m[2]/(-1+m[2 ])/(179945*m[1]+62080*m[2]-132104-60809*m[3])*F(m[1],-2+m[2],m[3])+4/437*( 968535968000*m[1]^3+488437191040*m[1]^2*m[2]-981895084800*m[1]^2*m[3]-\ 1635250735310*m[1]*m[2]^2+1350617347072*m[1]*m[2]*m[3]+331812821760*m[1]*m[3]^2 -582516878080*m[2]^3+1132445462414*m[2]^2*m[3]-512193766784*m[2]*m[3]^2-\ 37376616704*m[3]^3+668241914560*m[1]^2+13073338095042*m[1]*m[2]-6334206674432*m [1]*m[3]+5042219493104*m[2]^2-7101381415170*m[2]*m[3]+2064213535552*m[3]^2-\ 24716581982847*m[1]-12298402469136*m[2]+10351727243487*m[3]+5878846055736)/(-3+ m[2])/m[2]/(-1+m[2])/(-2+m[2])/(179945*m[1]+62080*m[2]-132104-60809*m[3])*F(m[1 ],m[2]-4,m[3])+1392/437*(79987711840*m[1]^2-32430388250*m[1]*m[2]+43790263552*m [1]*m[3]-20708522240*m[2]^2+44717246042*m[2]*m[3]-23932476512*m[3]^2+ 378920988797*m[1]+132206558032*m[2]-132368969117*m[3]-9722409000)/(-3+m[2])/m[2 ]/(-1+m[2])/(-2+m[2])/(179945*m[1]+62080*m[2]-132104-60809*m[3])*F(m[1],m[2]-6, m[3])+1084486320/437*(179945*m[1]+62080*m[2]-7944-60809*m[3])/(-3+m[2])/m[2]/(-\ 1+m[2])/(-2+m[2])/(179945*m[1]+62080*m[2]-132104-60809*m[3])*F(m[1],m[2]-8,m[3] ) F(m[1],m[2],m[3]) = -1/30*(109970624442*m[1]^2-187491839920*m[1]*m[2]+ 221620100988*m[1]*m[3]+54084735488*m[2]^2-113776858448*m[2]*m[3]+57001465290*m[ 3]^2-513368901669*m[1]+250709063184*m[2]-228952997295*m[3]+195218597475)/m[3]/( -1+m[3])/(3178893*m[1]-1169248*m[2]+962115*m[3]-1845525)*F(m[1],m[2],-2+m[3])+1 /60*(57770559721917*m[1]^3-63746824520736*m[1]^2*m[2]+34713305811177*m[1]^2*m[3 ]+23447107869696*m[1]*m[2]^2+95882974929280*m[1]*m[2]*m[3]-124769723470545*m[1] *m[3]^2-2874741194752*m[2]^3-39963618358272*m[2]^2*m[3]+82640712705440*m[2]*m[3 ]^2-39340627389525*m[3]^3+13929476152578*m[1]^2-435214170591200*m[1]*m[2]+ 996108111482904*m[1]*m[3]+158194274982912*m[2]^2-528746223853344*m[2]*m[3]+ 331529212887510*m[3]^2-1913569480337919*m[1]+803565248914304*m[2]-\ 763561934948505*m[3]+268347648848400)/(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/( 3178893*m[1]-1169248*m[2]+962115*m[3]-1845525)*F(m[1],m[2],m[3]-4)+29/30*( 220412596064682*m[1]^2+135931583942000*m[1]*m[2]-145239591219684*m[1]*m[3]-\ 79817152604672*m[2]^2+143635770208144*m[2]*m[3]-64147918889670*m[3]^2+ 1286950598974365*m[1]-465981234772400*m[2]+381685510628895*m[3]+34488647137395) /(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/(3178893*m[1]-1169248*m[2]+962115*m[3]-\ 1845525)*F(m[1],m[2],m[3]-6)+658222947/5*(3178893*m[1]-1169248*m[2]+962115*m[3] +78705)/(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/(3178893*m[1]-1169248*m[2]+962115*m[ 3]-1845525)*F(m[1],m[2],m[3]-8) Subject to the initial conditions F(0, 0, 0) = 1, F(0, 0, 1) = 0, F(0, 0, 2) = 1, F(0, 0, 3) = 0, F(0, 0, 4) = 1/2, F(0, 0, 5) = 0, F(0, 0, 6) = 1/6, F(0, 0, 7) = 0, F(0, 1, 0) = 0, F(0, 1, 1) = -56, F(0, 1, 2) = 0, F(0, 1, 3) = -56, F(0, 1, 4) = 0, F(0, 1, 5) = -28, F(0, 1, 6) = 0, F(0, 1, 7) = -28/3, F(0, 2, 0) = 1, F(0, 2, 1) = 0, F(0, 2, 2) = 1569, F(0, 2, 3) = 0, F(0, 2, 4) = 3137/2, F(0, 2, 5) = 0, F(0, 2, 6) = 4705/6, F(0, 2, 7) = 0, F(0, 3, 0) = 0, F(0, 3, 1) = -56, F(0, 3, 2) = 0, F(0, 3, 3) = -87976/3, F(0, 3, 4) = 0, F(0, 3, 5) = -87892/3, F(0, 3, 6) = 0, F(0, 3, 7) = -14644, F(0, 4, 0) = 1/2, F(0, 4, 1) = 0, F(0, 4, 2) = 3137/2, F(0, 4, 3) = 0, 4936067 F(0, 4, 4) = -------, F(0, 4, 5) = 0, F(0, 4, 6) = 1642219/4, 12 F(0, 4, 7) = 0, F(0, 5, 0) = 0, F(0, 5, 1) = -28, F(0, 5, 2) = 0, F(0, 5, 3) = -87892/3, F(0, 5, 4) = 0, F(0, 5, 5) = -23093574/5, F(0, 5, 6) = 0, F(0, 5, 7) = -23020354/5, F(0, 6, 0) = 1/6, F(0, 6, 1) = 0, F(0, 6, 2) = 4705/6, F(0, 6, 3) = 0, F(0, 6, 4) = 1642219/4, F(0, 6, 5) = 0, 2594691383 F(0, 6, 6) = ----------, F(0, 6, 7) = 0, F(0, 7, 0) = 0, F(0, 7, 1) = -28/3, 60 F(0, 7, 2) = 0, F(0, 7, 3) = -14644, F(0, 7, 4) = 0, -5209114498 F(0, 7, 5) = -23020354/5, F(0, 7, 6) = 0, F(0, 7, 7) = -----------, 15 F(1, 0, 0) = 0, F(1, 0, 1) = 21, F(1, 0, 2) = 0, F(1, 0, 3) = 21, F(1, 0, 4) = 0, F(1, 0, 5) = 21/2, F(1, 0, 6) = 0, F(1, 0, 7) = 7/2, F(1, 1, 0) = -8, F(1, 1, 1) = 0, F(1, 1, 2) = -1184, F(1, 1, 3) = 0, F(1, 1, 4) = -1180, F(1, 1, 5) = 0, F(1, 1, 6) = -1768/3, F(1, 1, 7) = 0, F(1, 2, 0) = 0, F(1, 2, 1) = 469, F(1, 2, 2) = 0, F(1, 2, 3) = 33397, F(1, 2, 4) = 0, F(1, 2, 5) = 66325/2, F(1, 2, 6) = 0, F(1, 2, 7) = 99253/6, F(1, 3, 0) = -8, F(1, 3, 1) = 0, F(1, 3, 2) = -13728, F(1, 3, 3) = 0, F(1, 3, 4) = -628380, F(1, 3, 5) = 0, F(1, 3, 6) = -1864552/3, F(1, 3, 7) = 0, F(1, 4, 0) = 0, F(1, 4, 1) = 917/2, F(1, 4, 2) = 0, 106469951 F(1, 4, 3) = 1605247/6, F(1, 4, 4) = 0, F(1, 4, 5) = ---------, 12 F(1, 4, 6) = 0, F(1, 4, 7) = 34955207/4, F(1, 5, 0) = -4, F(1, 5, 1) = 0, F(1, 5, 2) = -13136, F(1, 5, 3) = 0, F(1, 5, 4) = -11717866/3, F(1, 5, 5) = 0, F(1, 5, 6) = -501387244/5, F(1, 5, 7) = 0, F(1, 6, 0) = 0, F(1, 6, 1) = 455/2, F(1, 6, 2) = 0, F(1, 6, 3) = 1505077/6, F(1, 6, 4) = 0, 911427363 18899491009 F(1, 6, 5) = ---------, F(1, 6, 6) = 0, F(1, 6, 7) = -----------, 20 20 F(1, 7, 0) = -4/3, F(1, 7, 1) = 0, F(1, 7, 2) = -19408/3, F(1, 7, 3) = 0, -6639665068 F(1, 7, 4) = -3591962, F(1, 7, 5) = 0, F(1, 7, 6) = -----------, 15 F(1, 7, 7) = 0, F(2, 0, 0) = 1, F(2, 0, 1) = 0, F(2, 0, 2) = 443/2, 1325 F(2, 0, 3) = 0, F(2, 0, 4) = 221, F(2, 0, 5) = 0, F(2, 0, 6) = ----, 12 F(2, 0, 7) = 0, F(2, 1, 0) = 0, F(2, 1, 1) = -224, F(2, 1, 2) = 0, F(2, 1, 3) = -12572, F(2, 1, 4) = 0, F(2, 1, 5) = -12460, F(2, 1, 6) = 0, F(2, 1, 7) = -18634/3, F(2, 2, 0) = 33, F(2, 2, 1) = 0, F(2, 2, 2) = 22459/2, F(2, 2, 3) = 0, F(2, 2, 4) = 356957, F(2, 2, 5) = 0, F(2, 2, 6) = 1405391/4, F(2, 2, 7) = 0, F(2, 3, 0) = 0, F(2, 3, 1) = -2016, F(2, 3, 2) = 0, F(2, 3, 3) = -921172/3, F(2, 3, 4) = 0, F(2, 3, 5) = -20279812/3, F(2, 3, 6) = 0, F(2, 3, 7) = -19820234/3, F(2, 4, 0) = 65/2, F(2, 4, 1) = 0, F(2, 4, 2) = 245179/4, F(2, 4, 3) = 0, 2305387949 F(2, 4, 4) = 34404247/6, F(2, 4, 5) = 0, F(2, 4, 6) = ----------, 24 F(2, 4, 7) = 0, F(2, 5, 0) = 0, F(2, 5, 1) = -1904, F(2, 5, 2) = 0, -1216818442 F(2, 5, 3) = -1237390, F(2, 5, 4) = 0, F(2, 5, 5) = -----------, 15 F(2, 5, 6) = 0, F(2, 5, 7) = -5462362451/5, F(2, 6, 0) = 97/6, 668603 F(2, 6, 1) = 0, F(2, 6, 2) = ------, F(2, 6, 3) = 0, 12 110742482059 F(2, 6, 4) = 112009565/6, F(2, 6, 5) = 0, F(2, 6, 6) = ------------, 120 F(2, 6, 7) = 0, F(2, 7, 0) = 0, F(2, 7, 1) = -2800/3, F(2, 7, 2) = 0, F(2, 7, 3) = -3257870/3, F(2, 7, 4) = 0, F(2, 7, 5) = -1123025722/5, -131624070739 F(2, 7, 6) = 0, F(2, 7, 7) = -------------, F(3, 0, 0) = 0, F(3, 0, 1) = 21, 15 F(3, 0, 2) = 0, F(3, 0, 3) = 3129/2, F(3, 0, 4) = 0, F(3, 0, 5) = 1554, F(3, 0, 6) = 0, F(3, 0, 7) = 3101/4, F(3, 1, 0) = -8, F(3, 1, 1) = 0, F(3, 1, 2) = -2948, F(3, 1, 3) = 0, F(3, 1, 4) = -89380, F(3, 1, 5) = 0, F(3, 1, 6) = -263722/3, F(3, 1, 7) = 0, F(3, 2, 0) = 0, F(3, 2, 1) = 1141, F(3, 2, 2) = 0, F(3, 2, 3) = 268793/2, F(3, 2, 4) = 0, F(3, 2, 5) = 2554034, 29844311 F(3, 2, 6) = 0, F(3, 2, 7) = --------, F(3, 3, 0) = -280/3, F(3, 3, 1) = 0, 12 F(3, 3, 2) = -159628/3, F(3, 3, 3) = 0, F(3, 3, 4) = -10560620/3, F(3, 3, 5) = 0, F(3, 3, 6) = -438037502/9, F(3, 3, 7) = 0, F(3, 4, 0) = 0, 17114923 F(3, 4, 1) = 35455/6, F(3, 4, 2) = 0, F(3, 4, 3) = --------, F(3, 4, 4) = 0, 12 50101104389 F(3, 4, 5) = 64079785, F(3, 4, 6) = 0, F(3, 4, 7) = -----------, 72 F(3, 5, 0) = -268/3, F(3, 5, 1) = 0, F(3, 5, 2) = -185538, F(3, 5, 3) = 0, -40088450491 F(3, 5, 4) = -79669430/3, F(3, 5, 5) = 0, F(3, 5, 6) = ------------, 45 F(3, 5, 7) = 0, F(3, 6, 0) = 0, F(3, 6, 1) = 32053/6, F(3, 6, 2) = 0, 138850411 16982190743 F(3, 6, 3) = ---------, F(3, 6, 4) = 0, F(3, 6, 5) = -----------, 36 45 1200386535313 F(3, 6, 6) = 0, F(3, 6, 7) = -------------, F(3, 7, 0) = -44, 120 F(3, 7, 1) = 0, F(3, 7, 2) = -478274/3, F(3, 7, 3) = 0, F(3, 7, 4) = -538005302/9, F(3, 7, 5) = 0, F(3, 7, 6) = -21641381409/5, F(3, 7, 7) = 0, F(4, 0, 0) = 1/2, F(4, 0, 1) = 0, F(4, 0, 2) = 221, 197129 F(4, 0, 3) = 0, F(4, 0, 4) = 66593/8, F(4, 0, 5) = 0, F(4, 0, 6) = ------, 24 F(4, 0, 7) = 0, F(4, 1, 0) = 0, F(4, 1, 1) = -196, F(4, 1, 2) = 0, F(4, 1, 3) = -24892, F(4, 1, 4) = 0, F(4, 1, 5) = -478583, F(4, 1, 6) = 0, F(4, 1, 7) = -1398509/3, F(4, 2, 0) = 65/2, F(4, 2, 1) = 0, F(4, 2, 2) = 17501, F(4, 2, 3) = 0, F(4, 2, 4) = 8502561/8, F(4, 2, 5) = 0, 330244009 F(4, 2, 6) = ---------, F(4, 2, 7) = 0, F(4, 3, 0) = 0, F(4, 3, 1) = -3780, 24 F(4, 3, 2) = 0, F(4, 3, 3) = -2105012/3, F(4, 3, 4) = 0, F(4, 3, 5) = -80907365/3, F(4, 3, 6) = 0, F(4, 3, 7) = -791397901/3, 2435 F(4, 4, 0) = ----, F(4, 4, 1) = 0, F(4, 4, 2) = 1008535/6, F(4, 4, 3) = 0, 12 834970211 69377681371 F(4, 4, 4) = ---------, F(4, 4, 5) = 0, F(4, 4, 6) = -----------, 48 144 F(4, 4, 7) = 0, F(4, 5, 0) = 0, F(4, 5, 1) = -39718/3, F(4, 5, 2) = 0, -9244250617 F(4, 5, 3) = -13335770/3, F(4, 5, 4) = 0, F(4, 5, 5) = -----------, 30 -595342753187 F(4, 5, 6) = 0, F(4, 5, 7) = -------------, F(4, 6, 0) = 747/4, 90 F(4, 6, 1) = 0, F(4, 6, 2) = 2561885/6, F(4, 6, 3) = 0, 3969412897 3041561884301 F(4, 6, 4) = ----------, F(4, 6, 5) = 0, F(4, 6, 6) = -------------, 48 720 F(4, 6, 7) = 0, F(4, 7, 0) = 0, F(4, 7, 1) = -11382, F(4, 7, 2) = 0, -21256254565 F(4, 7, 3) = -81844826/9, F(4, 7, 4) = 0, F(4, 7, 5) = ------------, 18 -471173617593 F(4, 7, 6) = 0, F(4, 7, 7) = -------------, F(5, 0, 0) = 0, 10 F(5, 0, 1) = 21/2, F(5, 0, 2) = 0, F(5, 0, 3) = 1554, F(5, 0, 4) = 0, 1423317 1392307 F(5, 0, 5) = -------, F(5, 0, 6) = 0, F(5, 0, 7) = -------, F(5, 1, 0) = -4, 40 40 F(5, 1, 1) = 0, F(5, 1, 2) = -2356, F(5, 1, 3) = 0, F(5, 1, 4) = -153617, -30875302 F(5, 1, 5) = 0, F(5, 1, 6) = ---------, F(5, 1, 7) = 0, F(5, 2, 0) = 0, 15 F(5, 2, 1) = 1813/2, F(5, 2, 2) = 0, F(5, 2, 3) = 167090, F(5, 2, 4) = 0, 250047637 7144006009 F(5, 2, 5) = ---------, F(5, 2, 6) = 0, F(5, 2, 7) = ----------, 40 120 F(5, 3, 0) = -268/3, F(5, 3, 1) = 0, F(5, 3, 2) = -195484/3, F(5, 3, 3) = 0, -6963971026 F(5, 3, 4) = -18166835/3, F(5, 3, 5) = 0, F(5, 3, 6) = -----------, 45 111167 F(5, 3, 7) = 0, F(5, 4, 0) = 0, F(5, 4, 1) = ------, F(5, 4, 2) = 0, 12 34042176511 F(5, 4, 3) = 2399145, F(5, 4, 4) = 0, F(5, 4, 5) = -----------, 240 1961232520331 F(5, 4, 6) = 0, F(5, 4, 7) = -------------, F(5, 5, 0) = -1802/5, 720 -5981446 F(5, 5, 1) = 0, F(5, 5, 2) = --------, F(5, 5, 3) = 0, 15 -1713750271 -36320999923 F(5, 5, 4) = -----------, F(5, 5, 5) = 0, F(5, 5, 6) = ------------, 30 15 482083 F(5, 5, 7) = 0, F(5, 6, 0) = 0, F(5, 6, 1) = ------, F(5, 6, 2) = 0, 20 470183063 713738256959 F(5, 6, 3) = ---------, F(5, 6, 4) = 0, F(5, 6, 5) = ------------, 45 720 23275874839609 F(5, 6, 6) = 0, F(5, 6, 7) = --------------, F(5, 7, 0) = -1582/5, 720 F(5, 7, 1) = 0, F(5, 7, 2) = -3973718/5, F(5, 7, 3) = 0, -17497742591 F(5, 7, 4) = ------------, F(5, 7, 5) = 0, F(5, 7, 6) = -121051166887/9, 90 1325 F(5, 7, 7) = 0, F(6, 0, 0) = 1/6, F(6, 0, 1) = 0, F(6, 0, 2) = ----, 12 197129 91640261 F(6, 0, 3) = 0, F(6, 0, 4) = ------, F(6, 0, 5) = 0, F(6, 0, 6) = --------, 24 720 F(6, 0, 7) = 0, F(6, 1, 0) = 0, F(6, 1, 1) = -280/3, F(6, 1, 2) = 0, -11169466 F(6, 1, 3) = -55846/3, F(6, 1, 4) = 0, F(6, 1, 5) = ---------, 15 -666543353 F(6, 1, 6) = 0, F(6, 1, 7) = ----------, F(6, 2, 0) = 97/6, F(6, 2, 1) = 0, 90 27453865 F(6, 2, 2) = 48591/4, F(6, 2, 3) = 0, F(6, 2, 4) = --------, F(6, 2, 5) = 0, 24 21031460549 F(6, 2, 6) = -----------, F(6, 2, 7) = 0, F(6, 3, 0) = 0, 720 F(6, 3, 1) = -8344/3, F(6, 3, 2) = 0, F(6, 3, 3) = -6162674/9, -1733367118 F(6, 3, 4) = 0, F(6, 3, 5) = -----------, F(6, 3, 6) = 0, 45 -21268594067 F(6, 3, 7) = ------------, F(6, 4, 0) = 747/4, F(6, 4, 1) = 0, 30 4208045 3206816731 F(6, 4, 2) = -------, F(6, 4, 3) = 0, F(6, 4, 4) = ----------, 24 144 1243274075077 F(6, 4, 5) = 0, F(6, 4, 6) = -------------, F(6, 4, 7) = 0, F(6, 5, 0) = 0, 1440 -273432733 F(6, 5, 1) = -90132/5, F(6, 5, 2) = 0, F(6, 5, 3) = ----------, 45 F(6, 5, 4) = 0, F(6, 5, 5) = -2458873151/5, F(6, 5, 6) = 0, -2576121339001 32567 F(6, 5, 7) = --------------, F(6, 6, 0) = -----, F(6, 6, 1) = 0, 180 60 91005067 20203464937 F(6, 6, 2) = --------, F(6, 6, 3) = 0, F(6, 6, 4) = -----------, 120 144 35018891135173 F(6, 6, 5) = 0, F(6, 6, 6) = --------------, F(6, 6, 7) = 0, F(6, 7, 0) = 0, 4320 -555604 -296163637 F(6, 7, 1) = -------, F(6, 7, 2) = 0, F(6, 7, 3) = ----------, 15 15 F(6, 7, 4) = 0, F(6, 7, 5) = -21562556617/9, F(6, 7, 6) = 0, -57177745002553 F(6, 7, 7) = ---------------, F(7, 0, 0) = 0, F(7, 0, 1) = 7/2, 540 1392307 F(7, 0, 2) = 0, F(7, 0, 3) = 3101/4, F(7, 0, 4) = 0, F(7, 0, 5) = -------, 40 94027073 F(7, 0, 6) = 0, F(7, 0, 7) = --------, F(7, 1, 0) = -4/3, F(7, 1, 1) = 0, 240 F(7, 1, 2) = -3238/3, F(7, 1, 3) = 0, F(7, 1, 4) = -327371/3, -267070943 F(7, 1, 5) = 0, F(7, 1, 6) = ----------, F(7, 1, 7) = 0, F(7, 2, 0) = 0, 90 1265495 F(7, 2, 1) = 2485/6, F(7, 2, 2) = 0, F(7, 2, 3) = -------, F(7, 2, 4) = 0, 12 728255353 81435359747 F(7, 2, 5) = ---------, F(7, 2, 6) = 0, F(7, 2, 7) = -----------, 120 720 F(7, 3, 0) = -44, F(7, 3, 1) = 0, F(7, 3, 2) = -122534/3, F(7, 3, 3) = 0, -5794703029 F(7, 3, 4) = -15275179/3, F(7, 3, 5) = 0, F(7, 3, 6) = -----------, 30 F(7, 3, 7) = 0, F(7, 4, 0) = 0, F(7, 4, 1) = 25543/4, F(7, 4, 2) = 0, 143570693 21795423055 F(7, 4, 3) = ---------, F(7, 4, 4) = 0, F(7, 4, 5) = -----------, 72 144 6017260825667 F(7, 4, 6) = 0, F(7, 4, 7) = -------------, F(7, 5, 0) = -1582/5, 1440 F(7, 5, 1) = 0, F(7, 5, 2) = -1841973/5, F(7, 5, 3) = 0, -5400108581 -113539605011 F(7, 5, 4) = -----------, F(7, 5, 5) = 0, F(7, 5, 6) = -------------, 90 36 582337 F(7, 5, 7) = 0, F(7, 6, 0) = 0, F(7, 6, 1) = ------, F(7, 6, 2) = 0, 20 1464568721 182327491493 F(7, 6, 3) = ----------, F(7, 6, 4) = 0, F(7, 6, 5) = ------------, 120 144 23957500142891 -74626 F(7, 6, 6) = 0, F(7, 6, 7) = --------------, F(7, 7, 0) = ------, 480 105 -126515059 F(7, 7, 1) = 0, F(7, 7, 2) = ----------, F(7, 7, 3) = 0, 105 -57776476043 -76713852499033 F(7, 7, 4) = ------------, F(7, 7, 5) = 0, F(7, 7, 6) = ---------------, 210 3780 F(7, 7, 7) = 0 -------------------------------------------- By the way, it took, 7.041, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000, 3000) = 121013887184295629986121584548649840207590086419522618\ 125628366476235383378785777149902587496085557837435907082626924398627179\ 075374918094873406912731608955817708321265077463454989889255176869704598\ 173728239761789520666542664579228643280043688440469205590088020985107284\ 575195579503644931249420547453769404479167659707037674270777169476851608\ 400297794719961563698984938935162144914774974275762019356320171717972069\ 624134310106953266455670274797987954199274542089334903459675513366331222\ 401310101099693032614512736534263183066314982113277654646903079146398334\ 354404687003871362367405772693994383262812675138711969701827813271505552\ 669485100673620875052516942357837985637014809049283038224050155805225024\ 973890708802545003988524155561966983053680430240421227594135804473382887\ 340868848827531315139755765125132719223614013194645766892247802045055985\ 889039723275300118562130229507691756142203218328300017202379476441788937\ 504453454170705827522190105889442017566703891043773337791446873537146324\ 770189836878935752596435005456161087342697894516710918239632919988722951\ 249314484708093818688380027073110581044723684865304221993149283965626451\ 539985329755145570759612579116289782090588297746851453625256982640595733\ 117693162310637952178639597727041133468386665910942711423943066401177262\ 820685947154894065582890362870614342737177150014151731412314999978572073\ 403764636149027930964882541809151168206552534761448174285868402778120878\ 275622566041700599425958883955008925520050736340193629058315593799093167\ 944325788176464791667041551758774510567297040112174603781963033560273570\ 890837954452249843680914889900457633689573093874011914010242306036646276\ 099435093397469468718632493156958318227469388764297617279439481536179692\ 661864329829076901411858468644141579341144575584851028174839040349726113\ 056051570479808059419886389008329421953064487567064916563826479771581122\ 588132706782315679282367275392277292471761372238128400272987387825315722\ 889075199812960395440388176116492925379387972957167137370815937298235967\ 631084639350282072086606098478165714161028234360154442590505998290411249\ 593628920996514514922105011817112007545243248832237269696034428194873733\ 314897760117647223955504546545007896416751687232275336574894753537463313\ 416684619395094440521621514682156628917954498415645154090202731764170981\ 356311617705937195107623217323621866317118041514862979029949475142971032\ 761280001686443204883509959903867425603594560952025104933160826371486825\ 245135732815186655341205430463482981092446227450286395554681665815827302\ 112274096284998084680180618864389570835625706024977163098642116734806963\ 743703746851686486051881670838364005934333431491678116016853223089055305\ 127231516622688312526154336930766048239810939843483508122130636282036920\ 300102199033618306544999536074476942509468646026267917002308893515833897\ 956560396104759602817599136001255123951596353542792817235356088948582772\ 135127509952496568370559443465224352457833621540652621799498334525491755\ 418138084423233957142289939354387843880490379243269948279044921792575238\ 952182690319379751062263363221050209175933604021617581487216370819127874\ 030937530779038187996517493227325168743199269451100557169571860346472779\ 177741342271735384971575731793505616085619650845800320850276024926309498\ 498418545154182435347883677112986573007972982197938156262612470796323774\ 527178579084351509842542744838861237733120651349860123385071496386206648\ 791403255605644833067601649873253654502524490934625419232792350051526434\ 536804399442948298018240569599982235275513481667917378046428990229881461\ 614899209077663380601035926309774967630069441658840823652448366109863952\ 439747908738095177019753744014195113828619318289458960679640915659483796\ 614406728004657550020300987481341250391012688267292656103473874415518692\ 211190033022413855525737689453062579373092055288520439914756361525736818\ 422477265592945503957342208025479849868751295034364899208029653485818630\ 847273503315289613587873379679839343013627868012847005174007620191375805\ 552285933544266468054296134563736258775190440528630629541321168289745526\ 056685527020922955805890173178258553547107903517287138542187018457971074\ 754821305386606161366923174447106910386366743376740978574884139852588709\ 164993896990865217068177627971832643984996240789190182215845098885045683\ 928530046564556167656094423798654825143279106684960960749547444842775416\ 826570120258458151768259257712670857248873707356735144881597765796785084\ 114036636426612558430718250910813995116017569191726492351303413377443474\ 380931343259584470692520311804788638182654318558330699183927248923927838\ 388926908716519846025915328164207967471970157990034566002431213047022881\ 559668830965472700489353000530448603967204629687358164308732618114419838\ 774063785434287668073168585530478000257288120922687784793137122315019583\ 289192976460001216839094496688622185853083681100436275474576055262314602\ 620370381227030895678829270960107418236797092627003565870990551222082888\ / 3005384668734757113539207014601302532548763 / 775198587262182616572264\ / 258137166628787897990087759045829172236695815139114323320604064841577678\ 930183976675199700546441905746535848957541909113940840753111061344156618\ 127670344439085018684638176901032421676869890117762452131778939055934712\ 244085522181903677309717691526133668142951996756189429087056563586782368\ 960983300188824921783454201352282815641594591536217102017452439696976820\ 479710974924578884772429336283711479718529988148745844753499996529305884\ 867082806402696384722312224536903060525067263951196888699787844723999282\ 785295860235545546396716685343994967991084808898354353594717478833706718\ 397052657294102439615379892837103892788333146428518157997139232658800623\ 891185949031862912282943567705018697908680584286258119061884582380438883\ 934793319991075549149561535228518402283329095495475353646619076631213426\ 075632571625122803066024048727564388173129918741876484675098674671975369\ 992460397637638695679133330254492104595516686282120440579378027620178831\ 520114189222837165092044214173538232051822393940986329952968835041304689\ 424935575108653071055639331485792760632526842451413632179747613473907166\ 272064635242875642130538613239611188901065479727683770045920903202771666\ 144465865531949884837884612757332577531235747449643454103640791476810576\ 048537440457273341002087553926913519385484973139714254553047580785582624\ 609002785380411762832312115456339048838523653186651536752199786186130081\ 023728786604728131032431650947532363118449009187724683093989854466761157\ 048052480307428672172209224668017514119015409695800859071266407402296811\ 234952283764858032642879565795117358767585635368943495286795673409330932\ 582965018952162515225770819607740955140515327378445436546931989314967765\ 646623418974257400318690320332501440088153439698223216623753362216439315\ 856686236168187144118142809331931407118533708067157178715143713200338629\ 800888855930652923494200898530059836330589987107039513004952290547295159\ 694288480217381118025061041582635302662064954866955814151165983380813421\ 847913273155718840938864767308391097395046524721802896753697118711095955\ 433820166356987070648175271997412178081824046443303410282278332336566120\ 629426479318890501266910465113725650169514186857151993881077464133344101\ 931440983983929270362367970439852455056814134340356017894439720479791570\ 385643583294957512254493949006643050224146403947250311404952859720499684\ 400469967031263367519487105337796280851961436050158468940206515849345031\ 696473447974091555839875444361272896486438927491932963866077836258594326\ 058500001337068670003380625946119929775483911394093210046307362066507535\ 410157144091435392211623828116857613962749886701012473819451334670288446\ 922146832297434032110744942048361614538420581492290738479596388664162272\ 645330834367411723226936633838718861858372733715400492334751826927386568\ 664003646893474547288236479040600419343500238278643684717531599368811333\ 211799248319703145169345406217109286994012679934116191430561203008568193\ 114553114183587462412752129335816459176270333094115061934847530325932263\ 210002064643935457438082531702381702040264016607084824394762314218415882\ 835075487888524441010243163845670610501866377165310207959289905465708289\ 730807419485333820043328496623556445552488163956549922775110902781440846\ 186003273721183319694380562933467863656871571086020126876143405500887829\ 384102224381876708394567212422587434264835142290608348294894943635607242\ 044002240528621046209884087170675951802460202599205107795195469531901443\ 868229477989064005502799377792232811496518905055736070723703451813066545\ 379555330758397701235708453360111384591355981120671095692280187048230234\ 958644672907253377856696773892859866585398752762289095833234640289234057\ 329995537661199778790933236177170800846680096313652554110676517433461474\ 846069774953074358118398899105126216751003164443594144761862051898175856\ 350029435808830900636888031314317563722245760122308442508711443428373472\ 713200706383565122267363550851506685378384819745693448860719310480377313\ 693945009964917807941894983174693827213173774265758474977403845218043528\ 040684402062141593815118932880516259819328218457427594812892300185556235\ 859027049177231776800880021202575600060421722101344023976640552458944706\ 623708741349334736401701522334560234556318690846344824785324122107484014\ 611903535282284353218824205086260208529651492596260242708130108085027726\ 489960047890933293887135033203174270561231930779032917303856146620529562\ 963523335547145938571324957432127397172993526579707502909541986004809352\ 119951431733047995718585480800889583808289761675857727385489400720335454\ 217138369503772278612114650172719278788060259517482603852158800753494166\ 295891570438428162176112272779575857868442827091667839632209276838514264\ 845473838708052483937208450389948315345209797528205941693331085482712992\ 309526738793832776744908188384514462992107037590012275952058157721277512\ 124314215045993195131696310462079318012343933611577763770286485919617110\ 867543885196375581449280642301276486345858971899422549185455815276548457\ 500532244407844994031896794815215928688935042128409868703529608002060620\ 017305019291147876654683516215067816685388183828265582726872133430351489\ 946388196992344949139165050507223669219728251009401137835227007795771569\ 774851199154107313481640778135473324164682117848309569084681477300392108\ 010655070967888187263830794752044730973494459289932936344475203605946398\ 050648976028358225196745998664586285167936118297378527217816335972365296\ 030502781610899507652319660995887689114604695689926126720381428114712833\ 771769369240490860096799382921302967971745874958975548079669244703930334\ 292760070025608546613643121130519656523521353558675209613188453953543877\ 518555805870092638706941030140448476008366759015780662341745293916669050\ 860361170914208939688155424355505724851924212145006674900282270711991289\ 451960303180115180826981501522660339005154389956778022799856723712964875\ 098408547262565966572761634863313378344853247911500412911940761485691397\ 107126871567074720827130177898590174239241250353939430264177344649010547\ 504827160535614905967633223054894010548629512074007744043591660238081150\ 601778499962331638066519178037632013713335786194391842003767004264059672\ 077190952663556445065826156248195252398566573828971791604936384218955123\ 444678866711163928236497700788519596988035348199512406633538005819755516\ 423064290391494592573632336546924348547922643554189924232025795853802213\ 861775606942101686705856398943905159975640542246732993091498356841857382\ 528179933536927433232412690063930093645387835695064716943861065240294390\ 386760888497353500131901249175131184263563415559824182661380316082972884\ 358785438486637674538314411760508108885577735416515786098497424260202143\ 550241273270341430741764341968799296578529228951161040188893092698039973\ 742475019232624548576457311374744552986529802435282949636811770196541768\ 161883088428646935302826980568025240178064483536276054316501479624652823\ 976634998652190169750773633002555437904099798231533214778409879032215936\ 521527441955444364003299695122270298471158785011088438653079251436959716\ 130973299362367704621843404578101576464264886199344821210467319855041862\ 949530810499647737732801358127005816332223618236558626596723562954065206\ 664588338717976407861961037066348007305997465919869536640867880833673400\ 404795764605699368443729448201752904061135306794775521356607333532144572\ 967187660141219288486278754659868906569364412689809600556729953541882901\ 512455301074281826505260986367501103734803772011293350288206637535992456\ 155293281850866427681398399918317301440041100602184338219784353721945453\ 720321842342024171682035289971159163614730691570209007851073346855190647\ 807227930507264473028846391171989110784000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000 This took, 35.401, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 3, variables , x[1], x[2], x[3] 2 2 2 exp(x[1] - 53 x[1] x[2] + 43 x[1] x[3] + x[2] - 17 x[2] x[3] + x[3] ) By Shalosh B. Ekhad Theorem: Let , F(m[1], m[2], m[3]), be the coefficient of, m[1] m[2] m[3] x[1] x[2] x[3] , in the Taylor expansion (around the origing)\ of the multivariable function (that may also be viewed as a formal powe\ r series) of, x[1], x[2], x[3] 2 2 2 exp(x[1] - 53 x[1] x[2] + 43 x[1] x[3] + x[2] - 17 x[2] x[3] + x[3] ) The following PURE recurrence relations hold in each of the, 3, discrete variables, m[1], m[2], m[3] 2 F(m[1], m[2], m[3]) = -1/57 (1478198555052316000 m[1] - 7418948897589391625 m[1] m[2] + 7548689942871412233 m[1] m[3] 2 + 1967435256838801750 m[2] - 2740227597124809792 m[2] m[3] 2 + 696938289154525418 m[3] - 8604012350930187392 m[1] + 22263914381630116583 m[2] - 22435191316285508455 m[3] + 11185136452989130752) F(-2 + m[1], m[2], m[3])/(m[1] (-1 + m[1]) %1) - 3 2 5/228 (849971948561035748000 m[1] - 4336186122401222144375 m[1] m[2] 2 2 + 4055860243454935044599 m[1] m[3] + 1019549737879008883000 m[1] m[2] 2 + 361719043856586131008 m[1] m[2] m[3] - 1523850962783329910648 m[1] m[3] 3 2 + 2106453702241560546875 m[2] - 6685645584457855640625 m[2] m[3] 2 3 + 7073160739843790216625 m[2] m[3] - 2494379050342899402683 m[3] 2 - 8524704008280881747776 m[1] + 34157644762627456060008 m[1] m[2] 2 - 32826750731361668352360 m[1] m[3] - 4599183193531173840500 m[2] 2 + 914991832141523193472 m[2] m[3] + 4179766031941822329844 m[3] + 27140277586938840396288 m[1] - 67216882246842266709528 m[2] + 66303983637689247702552 m[3] - 26564724857655567129600) 25 F(m[1] - 4, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) - -- 38 2 (4668156415585111844000 m[1] - 26879032962073978666375 m[1] m[2] 2 + 18631386418960909419847 m[1] m[3] + 22559099066494128763625 m[2] 2 - 1274354380179835706752 m[2] m[3] - 23901823231427140808809 m[3] - 36168343345395300142528 m[1] + 141468867975472399320748 m[2] - 132732670140928965933868 m[3] + 48521664909511215810816) F(m[1] - 6, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) + 11661507000 ----------- (10129434836000 m[1] - 17495346541632 - 47993249178875 m[2] 19 + 50775046565243 m[3]) F(m[1] - 8, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) %1 := 10129434836000 m[1] - 37754216213632 - 47993249178875 m[2] + 50775046565243 m[3] 2 F(m[1], m[2], m[3]) = 1/369 (17477607406161763250 m[1] - 35770640967420157625 m[1] m[2] - 36574395985132931016 m[1] m[3] 2 + 17651166647868883500 m[2] + 38141750891967363929 m[2] m[3] 2 + 18938543490879732694 m[3] + 105157543028564881359 m[1] - 103246593130553211696 m[2] - 109409714647420644255 m[3] + 135714133559905762176) F(m[1], -2 + m[2], m[3])/(m[2] (-1 + m[2]) %1) + 3 2 5/1476 (1600806164802583984375 m[1] - 216434124959094541500 m[1] m[2] 2 2 - 5533110998915376703125 m[1] m[3] - 3651048396350212763875 m[1] m[2] + 1953530050213420205352 m[1] m[2] m[3] 2 3 + 6374979057316387981125 m[1] m[3] + 2264344368194130442500 m[2] 2 2 + 2998674096838253306995 m[2] m[3] - 1963455622333807124844 m[2] m[3] 3 2 - 2448312711672754186663 m[3] - 1934654493924243073500 m[1] + 27719563844734621739352 m[1] m[2] - 633773706322536725232 m[1] m[3] 2 - 22780432439253164807880 m[2] - 24297470741482761527352 m[2] m[3] 2 + 3298363556951969867340 m[3] - 52031289611966363342232 m[1] + 73011050559758477500992 m[2] + 48609651944763284082072 m[3] - 72483575851420657693056) F(m[1], m[2] - 4, m[3])/((-3 + m[2]) m[2] 25 2 (-1 + m[2]) (-2 + m[2]) %1) + --- (2738345002462192456125 m[1] 246 - 4717523544945868559375 m[1] m[2] - 350367812125111535664 m[1] m[3] 2 + 2029145760581791330500 m[2] + 487186386765913381607 m[2] m[3] 2 - 3231344387320384136685 m[3] + 19182837537058085274836 m[1] - 16149642644506174263768 m[2] - 13072120518739215848996 m[3] + 22671449904055084028064) F(m[1], m[2] - 6, m[3])/((-3 + m[2]) m[2] 600457000 (-1 + m[2]) (-2 + m[2]) %1) + --------- (36472621768375 m[1] 41 - 42021965578567 m[3] - 30282313120500 m[2] + 55259700968808) F(m[1], m[2] - 8, m[3])/((-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1) %1 := 36472621768375 m[1] - 30282313120500 m[2] + 115824327209808 - 42021965578567 m[3] 2 F(m[1], m[2], m[3]) = 1/561 (12380432434082963026 m[1] - 25061369982736321416 m[1] m[2] - 25906516178689670457 m[1] m[3] 2 + 12609563052440510198 m[2] + 26796523332472173785 m[2] m[3] 2 + 12421022790486545612 m[3] + 76992653002412207503 m[1] - 78339367617042497759 m[2] - 73440402104578361008 m[3] + 97147828463393874816) F(m[1], m[2], -2 + m[3])/(m[3] (-1 + m[3]) %1) + 3 2 5/2244 (1344919368300938040023 m[1] - 4393966131362987612853 m[1] m[2] 2 2 - 447222761744133702556 m[1] m[3] + 4785153870836593586661 m[1] m[2] + 1706390631829118631656 m[1] m[2] m[3] 2 3 - 2315404815733076379971 m[1] m[3] - 1737056141917330995719 m[2] 2 2 - 1327909325502208146508 m[2] m[3] + 1976740053870102187091 m[2] m[3] 3 2 + 1416294912197173495396 m[3] - 269981561343251558812 m[1] - 1975064044300062427504 m[1] m[2] + 17919306046949572229592 m[1] m[3] 2 + 2471094318920416321676 m[2] - 16034124742981708804536 m[2] m[3] 2 - 14259222773543761387400 m[3] - 34428208534580886239640 m[1] + 32257737005463971463576 m[2] + 45819028944748155170880 m[3] - 45774834139827797726592) F(m[1], m[2], m[3] - 4)/((-3 + m[3]) m[3] 25 2 (-1 + m[3]) (-2 + m[3]) %1) + --- (1389790585550974061917 m[1] 374 - 92935541943812750000 m[1] m[2] - 2202736569291451720975 m[1] m[3] 2 - 1547058340081300648973 m[2] + 216027649025813312551 m[2] m[3] 2 + 869529498449665063972 m[3] + 9223476419252965404052 m[1] - 5922206302295742712420 m[2] - 7035206788081272661016 m[3] + 10167320203064806213536) F(m[1], m[2], m[3] - 6)/((-3 + m[3]) m[3] 1184859000 (-1 + m[3]) (-2 + m[3]) %1) + ---------- (-29814211663271 m[2] 187 - 20366906650324 m[3] + 38464538257704 + 27376891071383 m[1]) F(m[1], m[2], m[3] - 8)/((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) %1 := 27376891071383 m[1] - 29814211663271 m[2] - 20366906650324 m[3] + 79198351558352 and in Maple notation F(m[1],m[2],m[3]) = -1/57*(1478198555052316000*m[1]^2-7418948897589391625*m[1]* m[2]+7548689942871412233*m[1]*m[3]+1967435256838801750*m[2]^2-\ 2740227597124809792*m[2]*m[3]+696938289154525418*m[3]^2-8604012350930187392*m[1 ]+22263914381630116583*m[2]-22435191316285508455*m[3]+11185136452989130752)/m[1 ]/(-1+m[1])/(10129434836000*m[1]-37754216213632-47993249178875*m[2]+ 50775046565243*m[3])*F(-2+m[1],m[2],m[3])-5/228*(849971948561035748000*m[1]^3-\ 4336186122401222144375*m[1]^2*m[2]+4055860243454935044599*m[1]^2*m[3]+ 1019549737879008883000*m[1]*m[2]^2+361719043856586131008*m[1]*m[2]*m[3]-\ 1523850962783329910648*m[1]*m[3]^2+2106453702241560546875*m[2]^3-\ 6685645584457855640625*m[2]^2*m[3]+7073160739843790216625*m[2]*m[3]^2-\ 2494379050342899402683*m[3]^3-8524704008280881747776*m[1]^2+ 34157644762627456060008*m[1]*m[2]-32826750731361668352360*m[1]*m[3]-\ 4599183193531173840500*m[2]^2+914991832141523193472*m[2]*m[3]+ 4179766031941822329844*m[3]^2+27140277586938840396288*m[1]-\ 67216882246842266709528*m[2]+66303983637689247702552*m[3]-\ 26564724857655567129600)/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[1])/(10129434836000*m[1 ]-37754216213632-47993249178875*m[2]+50775046565243*m[3])*F(m[1]-4,m[2],m[3])-\ 25/38*(4668156415585111844000*m[1]^2-26879032962073978666375*m[1]*m[2]+ 18631386418960909419847*m[1]*m[3]+22559099066494128763625*m[2]^2-\ 1274354380179835706752*m[2]*m[3]-23901823231427140808809*m[3]^2-\ 36168343345395300142528*m[1]+141468867975472399320748*m[2]-\ 132732670140928965933868*m[3]+48521664909511215810816)/(-3+m[1])/m[1]/(-1+m[1]) /(-2+m[1])/(10129434836000*m[1]-37754216213632-47993249178875*m[2]+ 50775046565243*m[3])*F(m[1]-6,m[2],m[3])+11661507000/19*(10129434836000*m[1]-\ 17495346541632-47993249178875*m[2]+50775046565243*m[3])/(-3+m[1])/m[1]/(-1+m[1] )/(-2+m[1])/(10129434836000*m[1]-37754216213632-47993249178875*m[2]+ 50775046565243*m[3])*F(m[1]-8,m[2],m[3]) F(m[1],m[2],m[3]) = 1/369*(17477607406161763250*m[1]^2-35770640967420157625*m[1 ]*m[2]-36574395985132931016*m[1]*m[3]+17651166647868883500*m[2]^2+ 38141750891967363929*m[2]*m[3]+18938543490879732694*m[3]^2+ 105157543028564881359*m[1]-103246593130553211696*m[2]-109409714647420644255*m[3 ]+135714133559905762176)/m[2]/(-1+m[2])/(36472621768375*m[1]-30282313120500*m[2 ]+115824327209808-42021965578567*m[3])*F(m[1],-2+m[2],m[3])+5/1476*( 1600806164802583984375*m[1]^3-216434124959094541500*m[1]^2*m[2]-\ 5533110998915376703125*m[1]^2*m[3]-3651048396350212763875*m[1]*m[2]^2+ 1953530050213420205352*m[1]*m[2]*m[3]+6374979057316387981125*m[1]*m[3]^2+ 2264344368194130442500*m[2]^3+2998674096838253306995*m[2]^2*m[3]-\ 1963455622333807124844*m[2]*m[3]^2-2448312711672754186663*m[3]^3-\ 1934654493924243073500*m[1]^2+27719563844734621739352*m[1]*m[2]-\ 633773706322536725232*m[1]*m[3]-22780432439253164807880*m[2]^2-\ 24297470741482761527352*m[2]*m[3]+3298363556951969867340*m[3]^2-\ 52031289611966363342232*m[1]+73011050559758477500992*m[2]+ 48609651944763284082072*m[3]-72483575851420657693056)/(-3+m[2])/m[2]/(-1+m[2])/ (-2+m[2])/(36472621768375*m[1]-30282313120500*m[2]+115824327209808-\ 42021965578567*m[3])*F(m[1],m[2]-4,m[3])+25/246*(2738345002462192456125*m[1]^2-\ 4717523544945868559375*m[1]*m[2]-350367812125111535664*m[1]*m[3]+ 2029145760581791330500*m[2]^2+487186386765913381607*m[2]*m[3]-\ 3231344387320384136685*m[3]^2+19182837537058085274836*m[1]-\ 16149642644506174263768*m[2]-13072120518739215848996*m[3]+ 22671449904055084028064)/(-3+m[2])/m[2]/(-1+m[2])/(-2+m[2])/(36472621768375*m[1 ]-30282313120500*m[2]+115824327209808-42021965578567*m[3])*F(m[1],m[2]-6,m[3])+ 600457000/41*(36472621768375*m[1]-42021965578567*m[3]-30282313120500*m[2]+ 55259700968808)/(-3+m[2])/m[2]/(-1+m[2])/(-2+m[2])/(36472621768375*m[1]-\ 30282313120500*m[2]+115824327209808-42021965578567*m[3])*F(m[1],m[2]-8,m[3]) F(m[1],m[2],m[3]) = 1/561*(12380432434082963026*m[1]^2-25061369982736321416*m[1 ]*m[2]-25906516178689670457*m[1]*m[3]+12609563052440510198*m[2]^2+ 26796523332472173785*m[2]*m[3]+12421022790486545612*m[3]^2+76992653002412207503 *m[1]-78339367617042497759*m[2]-73440402104578361008*m[3]+97147828463393874816) /m[3]/(-1+m[3])/(27376891071383*m[1]-29814211663271*m[2]-20366906650324*m[3]+ 79198351558352)*F(m[1],m[2],-2+m[3])+5/2244*(1344919368300938040023*m[1]^3-\ 4393966131362987612853*m[1]^2*m[2]-447222761744133702556*m[1]^2*m[3]+ 4785153870836593586661*m[1]*m[2]^2+1706390631829118631656*m[1]*m[2]*m[3]-\ 2315404815733076379971*m[1]*m[3]^2-1737056141917330995719*m[2]^3-\ 1327909325502208146508*m[2]^2*m[3]+1976740053870102187091*m[2]*m[3]^2+ 1416294912197173495396*m[3]^3-269981561343251558812*m[1]^2-\ 1975064044300062427504*m[1]*m[2]+17919306046949572229592*m[1]*m[3]+ 2471094318920416321676*m[2]^2-16034124742981708804536*m[2]*m[3]-\ 14259222773543761387400*m[3]^2-34428208534580886239640*m[1]+ 32257737005463971463576*m[2]+45819028944748155170880*m[3]-\ 45774834139827797726592)/(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/(27376891071383*m[1 ]-29814211663271*m[2]-20366906650324*m[3]+79198351558352)*F(m[1],m[2],m[3]-4)+ 25/374*(1389790585550974061917*m[1]^2-92935541943812750000*m[1]*m[2]-\ 2202736569291451720975*m[1]*m[3]-1547058340081300648973*m[2]^2+ 216027649025813312551*m[2]*m[3]+869529498449665063972*m[3]^2+ 9223476419252965404052*m[1]-5922206302295742712420*m[2]-7035206788081272661016* m[3]+10167320203064806213536)/(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/( 27376891071383*m[1]-29814211663271*m[2]-20366906650324*m[3]+79198351558352)*F(m [1],m[2],m[3]-6)+1184859000/187*(-29814211663271*m[2]-20366906650324*m[3]+ 38464538257704+27376891071383*m[1])/(-3+m[3])/m[3]/(-1+m[3])/(-2+m[3])/( 27376891071383*m[1]-29814211663271*m[2]-20366906650324*m[3]+79198351558352)*F(m [1],m[2],m[3]-8) Subject to the initial conditions F(0, 0, 0) = 1, F(0, 0, 1) = 0, F(0, 0, 2) = 1, F(0, 0, 3) = 0, F(0, 0, 4) = 1/2, F(0, 0, 5) = 0, F(0, 0, 6) = 1/6, F(0, 0, 7) = 0, F(0, 1, 0) = 0, F(0, 1, 1) = -17, F(0, 1, 2) = 0, F(0, 1, 3) = -17, F(0, 1, 4) = 0, F(0, 1, 5) = -17/2, F(0, 1, 6) = 0, F(0, 1, 7) = -17/6, F(0, 2, 0) = 1, F(0, 2, 1) = 0, F(0, 2, 2) = 291/2, F(0, 2, 3) = 0, 869 F(0, 2, 4) = 145, F(0, 2, 5) = 0, F(0, 2, 6) = ---, F(0, 2, 7) = 0, 12 F(0, 3, 0) = 0, F(0, 3, 1) = -17, F(0, 3, 2) = 0, F(0, 3, 3) = -5015/6, F(0, 3, 4) = 0, F(0, 3, 5) = -2482/3, F(0, 3, 6) = 0, F(0, 3, 7) = -1649/4, F(0, 4, 0) = 1/2, F(0, 4, 1) = 0, F(0, 4, 2) = 145, F(0, 4, 3) = 0, 86995 F(0, 4, 4) = -----, F(0, 4, 5) = 0, F(0, 4, 6) = 28419/8, F(0, 4, 7) = 0, 24 F(0, 5, 0) = 0, F(0, 5, 1) = -17/2, F(0, 5, 2) = 0, F(0, 5, 3) = -2482/3, -506209 -489719 F(0, 5, 4) = 0, F(0, 5, 5) = -------, F(0, 5, 6) = 0, F(0, 5, 7) = -------, 40 40 869 F(0, 6, 0) = 1/6, F(0, 6, 1) = 0, F(0, 6, 2) = ---, F(0, 6, 3) = 0, 12 8889743 F(0, 6, 4) = 28419/8, F(0, 6, 5) = 0, F(0, 6, 6) = -------, F(0, 6, 7) = 0, 240 F(0, 7, 0) = 0, F(0, 7, 1) = -17/6, F(0, 7, 2) = 0, F(0, 7, 3) = -1649/4, -489719 F(0, 7, 4) = 0, F(0, 7, 5) = -------, F(0, 7, 6) = 0, 40 -157002259 F(0, 7, 7) = ----------, F(1, 0, 0) = 0, F(1, 0, 1) = 43, F(1, 0, 2) = 0, 1680 F(1, 0, 3) = 43, F(1, 0, 4) = 0, F(1, 0, 5) = 43/2, F(1, 0, 6) = 0, F(1, 0, 7) = 43/6, F(1, 1, 0) = -53, F(1, 1, 1) = 0, F(1, 1, 2) = -784, F(1, 1, 3) = 0, F(1, 1, 4) = -1515/2, F(1, 1, 5) = 0, F(1, 1, 6) = -1123/3, F(1, 1, 7) = 0, F(1, 2, 0) = 0, F(1, 2, 1) = 944, F(1, 2, 2) = 0, F(1, 2, 3) = 14315/2, F(1, 2, 4) = 0, F(1, 2, 5) = 13371/2, F(1, 2, 6) = 0, 39169 F(1, 2, 7) = -----, F(1, 3, 0) = -53, F(1, 3, 1) = 0, F(1, 3, 2) = -16885/2, 12 F(1, 3, 3) = 0, F(1, 3, 4) = -261755/6, F(1, 3, 5) = 0, -472961 F(1, 3, 6) = -------, F(1, 3, 7) = 0, F(1, 4, 0) = 0, F(1, 4, 1) = 1845/2, 12 4793153 F(1, 4, 2) = 0, F(1, 4, 3) = 303205/6, F(1, 4, 4) = 0, F(1, 4, 5) = -------, 24 F(1, 4, 6) = 0, F(1, 4, 7) = 1396811/8, F(1, 5, 0) = -53/2, F(1, 5, 1) = 0, -5464543 F(1, 5, 2) = -16101/2, F(1, 5, 3) = 0, F(1, 5, 4) = --------, 24 -14649011 F(1, 5, 5) = 0, F(1, 5, 6) = ---------, F(1, 5, 7) = 0, F(1, 6, 0) = 0, 20 563551 F(1, 6, 1) = 1373/3, F(1, 6, 2) = 0, F(1, 6, 3) = ------, F(1, 6, 4) = 0, 12 16469581 537989591 F(1, 6, 5) = --------, F(1, 6, 6) = 0, F(1, 6, 7) = ---------, 20 240 -47519 F(1, 7, 0) = -53/6, F(1, 7, 1) = 0, F(1, 7, 2) = ------, F(1, 7, 3) = 0, 12 -597503881 F(1, 7, 4) = -1648021/8, F(1, 7, 5) = 0, F(1, 7, 6) = ----------, 240 F(1, 7, 7) = 0, F(2, 0, 0) = 1, F(2, 0, 1) = 0, F(2, 0, 2) = 1851/2, 5549 F(2, 0, 3) = 0, F(2, 0, 4) = 925, F(2, 0, 5) = 0, F(2, 0, 6) = ----, 12 F(2, 0, 7) = 0, F(2, 1, 0) = 0, F(2, 1, 1) = -2296, F(2, 1, 2) = 0, F(2, 1, 3) = -36025/2, F(2, 1, 4) = 0, F(2, 1, 5) = -33729/2, -98891 F(2, 1, 6) = 0, F(2, 1, 7) = ------, F(2, 2, 0) = 2811/2, F(2, 2, 1) = 0, 12 F(2, 2, 2) = 82435/2, F(2, 2, 3) = 0, F(2, 2, 4) = 174105, F(2, 2, 5) = 0, F(2, 2, 6) = 307461/2, F(2, 2, 7) = 0, F(2, 3, 0) = 0, F(2, 3, 1) = -52345/2, F(2, 3, 2) = 0, F(2, 3, 3) = -1116070/3, F(2, 3, 4) = 0, F(2, 3, 5) = -2231897/2, F(2, 3, 6) = 0, -11211587 F(2, 3, 7) = ---------, F(2, 4, 0) = 1405, F(2, 4, 1) = 0, 12 53908655 F(2, 4, 2) = 243705, F(2, 4, 3) = 0, F(2, 4, 4) = --------, F(2, 4, 5) = 0, 24 256409959 F(2, 4, 6) = ---------, F(2, 4, 7) = 0, F(2, 5, 0) = 0, 48 F(2, 5, 1) = -50049/2, F(2, 5, 2) = 0, F(2, 5, 3) = -3026137/2, -306012443 F(2, 5, 4) = 0, F(2, 5, 5) = ----------, F(2, 5, 6) = 0, 30 -1630949299 8429 F(2, 5, 7) = -----------, F(2, 6, 0) = ----, F(2, 6, 1) = 0, 80 12 338256679 F(2, 6, 2) = 446501/2, F(2, 6, 3) = 0, F(2, 6, 4) = ---------, 48 8916427139 F(2, 6, 5) = 0, F(2, 6, 6) = ----------, F(2, 6, 7) = 0, F(2, 7, 0) = 0, 240 -147851 -15960707 F(2, 7, 1) = -------, F(2, 7, 2) = 0, F(2, 7, 3) = ---------, 12 12 -2101079539 F(2, 7, 4) = 0, F(2, 7, 5) = -----------, F(2, 7, 6) = 0, 80 -4746960137 F(2, 7, 7) = -----------, F(3, 0, 0) = 0, F(3, 0, 1) = 43, F(3, 0, 2) = 0, 42 F(3, 0, 3) = 79765/6, F(3, 0, 4) = 0, F(3, 0, 5) = 39818/3, F(3, 0, 6) = 0, F(3, 0, 7) = 26531/4, F(3, 1, 0) = -53, F(3, 1, 1) = 0, F(3, 1, 2) = -99565/2, F(3, 1, 3) = 0, F(3, 1, 4) = -1650155/6, -3001721 F(3, 1, 5) = 0, F(3, 1, 6) = --------, F(3, 1, 7) = 0, F(3, 2, 0) = 0, 12 F(3, 2, 1) = 122675/2, F(3, 2, 2) = 0, F(3, 2, 3) = 2741330/3, F(3, 2, 4) = 0, F(3, 2, 5) = 5595803/2, F(3, 2, 6) = 0, 28214833 F(3, 2, 7) = --------, F(3, 3, 0) = -149195/6, F(3, 3, 1) = 0, 12 F(3, 3, 2) = -3326830/3, F(3, 3, 3) = 0, F(3, 3, 4) = -25311820/3, -337273973 F(3, 3, 5) = 0, F(3, 3, 6) = ----------, F(3, 3, 7) = 0, F(3, 4, 0) = 0, 18 F(3, 4, 1) = 2898805/6, F(3, 4, 2) = 0, F(3, 4, 3) = 30297620/3, 1249048321 F(3, 4, 4) = 0, F(3, 4, 5) = ----------, F(3, 4, 6) = 0, 24 13419246413 F(3, 4, 7) = -----------, F(3, 5, 0) = -74518/3, F(3, 5, 1) = 0, 144 -1476524831 F(3, 5, 2) = -9339013/2, F(3, 5, 3) = 0, F(3, 5, 4) = -----------, 24 -173568495791 F(3, 5, 5) = 0, F(3, 5, 6) = -------------, F(3, 5, 7) = 0, F(3, 6, 0) = 0, 720 5429671 539317963 F(3, 6, 1) = -------, F(3, 6, 2) = 0, F(3, 6, 3) = ---------, 12 18 202869731761 F(3, 6, 4) = 0, F(3, 6, 5) = ------------, F(3, 6, 6) = 0, 720 F(3, 6, 7) = 3576684915/4, F(3, 7, 0) = -49661/4, F(3, 7, 1) = 0, -49479983 -20692621843 F(3, 7, 2) = ---------, F(3, 7, 3) = 0, F(3, 7, 4) = ------------, 12 144 F(3, 7, 5) = 0, F(3, 7, 6) = -4137805037/4, F(3, 7, 7) = 0, F(4, 0, 0) = 1/2, F(4, 0, 1) = 0, F(4, 0, 2) = 925, F(4, 0, 3) = 0, 3440995 F(4, 0, 4) = -------, F(4, 0, 5) = 0, F(4, 0, 6) = 1143299/8, 24 F(4, 0, 7) = 0, F(4, 1, 0) = 0, F(4, 1, 1) = -4575/2, F(4, 1, 2) = 0, -75379747 F(4, 1, 3) = -4321895/6, F(4, 1, 4) = 0, F(4, 1, 5) = ---------, 24 F(4, 1, 6) = 0, F(4, 1, 7) = -22248369/8, F(4, 2, 0) = 1405, F(4, 2, 1) = 0, 325301855 F(4, 2, 2) = 1339605, F(4, 2, 3) = 0, F(4, 2, 4) = ---------, 24 1606497919 F(4, 2, 5) = 0, F(4, 2, 6) = ----------, F(4, 2, 7) = 0, F(4, 3, 0) = 0, 48 F(4, 3, 1) = -6558695/6, F(4, 3, 2) = 0, F(4, 3, 3) = -72644080/3, -3079940579 F(4, 3, 4) = 0, F(4, 3, 5) = -----------, F(4, 3, 6) = 0, 24 -33558989647 7924195 F(4, 3, 7) = ------------, F(4, 4, 0) = -------, F(4, 4, 1) = 0, 144 24 480217055 3543735045 F(4, 4, 2) = ---------, F(4, 4, 3) = 0, F(4, 4, 4) = ----------, 24 16 116699272781 F(4, 4, 5) = 0, F(4, 4, 6) = ------------, F(4, 4, 7) = 0, F(4, 5, 0) = 0, 144 -160345507 -4434437219 F(4, 5, 1) = ----------, F(4, 5, 2) = 0, F(4, 5, 3) = -----------, 24 24 -108483123879 F(4, 5, 4) = 0, F(4, 5, 5) = -------------, F(4, 5, 6) = 0, 80 -138111283823 F(4, 5, 7) = -------------, F(4, 6, 0) = 2635779/8, F(4, 6, 1) = 0, 36 3208639999 164272453901 F(4, 6, 2) = ----------, F(4, 6, 3) = 0, F(4, 6, 4) = ------------, 48 144 449400682637 F(4, 6, 5) = 0, F(4, 6, 6) = ------------, F(4, 6, 7) = 0, F(4, 7, 0) = 0, 72 -63646386127 F(4, 7, 1) = -49079089/8, F(4, 7, 2) = 0, F(4, 7, 3) = ------------, 144 -190485361391 F(4, 7, 4) = 0, F(4, 7, 5) = -------------, F(4, 7, 6) = 0, 36 -3868138049677 F(4, 7, 7) = --------------, F(5, 0, 0) = 0, F(5, 0, 1) = 43/2, 168 49533291 F(5, 0, 2) = 0, F(5, 0, 3) = 39818/3, F(5, 0, 4) = 0, F(5, 0, 5) = --------, 40 49267981 F(5, 0, 6) = 0, F(5, 0, 7) = --------, F(5, 1, 0) = -53/2, F(5, 1, 1) = 0, 40 -187787983 F(5, 1, 2) = -98781/2, F(5, 1, 3) = 0, F(5, 1, 4) = ----------, 24 -572520091 F(5, 1, 5) = 0, F(5, 1, 6) = ----------, F(5, 1, 7) = 0, F(5, 2, 0) = 0, 20 F(5, 2, 1) = 121731/2, F(5, 2, 2) = 0, F(5, 2, 3) = 39042923/2, 4529351407 F(5, 2, 4) = 0, F(5, 2, 5) = ----------, F(5, 2, 6) = 0, 30 25460036841 F(5, 2, 7) = -----------, F(5, 3, 0) = -74518/3, F(5, 3, 1) = 0, 80 -8527110191 F(5, 3, 2) = -48088373/2, F(5, 3, 3) = 0, F(5, 3, 4) = -----------, 24 -1055454222071 F(5, 3, 5) = 0, F(5, 3, 6) = --------------, F(5, 3, 7) = 0, F(5, 4, 0) = 0, 720 350874833 10387037041 F(5, 4, 1) = ---------, F(5, 4, 2) = 0, F(5, 4, 3) = -----------, 24 24 262870571821 F(5, 4, 4) = 0, F(5, 4, 5) = ------------, F(5, 4, 6) = 0, 80 341176683685 -140391541 F(5, 4, 7) = ------------, F(5, 5, 0) = ----------, F(5, 5, 1) = 0, 36 40 -8142624257 -316907190771 F(5, 5, 2) = -----------, F(5, 5, 3) = 0, F(5, 5, 4) = -------------, 30 80 -610461776831 F(5, 5, 5) = 0, F(5, 5, 6) = -------------, F(5, 5, 7) = 0, F(5, 6, 0) = 0, 30 1476674341 1832694682921 F(5, 6, 1) = ----------, F(5, 6, 2) = 0, F(5, 6, 3) = -------------, 20 720 728923059521 F(5, 6, 4) = 0, F(5, 6, 5) = ------------, F(5, 6, 6) = 0, 30 34092786095569 -139894931 F(5, 6, 7) = --------------, F(5, 7, 0) = ----------, F(5, 7, 1) = 0, 360 40 -61038721591 -574230995195 F(5, 7, 2) = ------------, F(5, 7, 3) = 0, F(5, 7, 4) = -------------, 80 36 -40348938240719 F(5, 7, 5) = 0, F(5, 7, 6) = ---------------, F(5, 7, 7) = 0, 360 5549 F(6, 0, 0) = 1/6, F(6, 0, 1) = 0, F(6, 0, 2) = ----, F(6, 0, 3) = 0, 12 2141364503 F(6, 0, 4) = 1143299/8, F(6, 0, 5) = 0, F(6, 0, 6) = ----------, 240 F(6, 0, 7) = 0, F(6, 1, 0) = 0, F(6, 1, 1) = -3427/3, F(6, 1, 2) = 0, -8535749 -1361222419 F(6, 1, 3) = --------, F(6, 1, 4) = 0, F(6, 1, 5) = -----------, 12 20 -52070414509 8429 F(6, 1, 6) = 0, F(6, 1, 7) = ------------, F(6, 2, 0) = ----, 240 12 F(6, 2, 1) = 0, F(6, 2, 2) = 2638041/2, F(6, 2, 3) = 0, 10249838359 323047440179 F(6, 2, 4) = -----------, F(6, 2, 5) = 0, F(6, 2, 6) = ------------, 48 240 -12960389 F(6, 2, 7) = 0, F(6, 3, 0) = 0, F(6, 3, 1) = ---------, F(6, 3, 2) = 0, 12 -6350900597 -2824350595139 F(6, 3, 3) = -----------, F(6, 3, 4) = 0, F(6, 3, 5) = --------------, 18 720 -160734886163 F(6, 3, 6) = 0, F(6, 3, 7) = -------------, F(6, 4, 0) = 2635779/8, 12 15544085719 F(6, 4, 1) = 0, F(6, 4, 2) = -----------, F(6, 4, 3) = 0, 48 909210638021 2647282963205 F(6, 4, 4) = ------------, F(6, 4, 5) = 0, F(6, 4, 6) = -------------, 144 72 -3130438739 F(6, 4, 7) = 0, F(6, 5, 0) = 0, F(6, 5, 1) = -----------, F(6, 5, 2) = 0, 20 -4197440560259 -1736007234839 F(6, 5, 3) = --------------, F(6, 5, 4) = 0, F(6, 5, 5) = --------------, 720 30 -83097357479051 7467109463 F(6, 5, 6) = 0, F(6, 5, 7) = ---------------, F(6, 6, 0) = ----------, 360 240 707753840819 F(6, 6, 1) = 0, F(6, 6, 2) = ------------, F(6, 6, 3) = 0, 240 3860222098373 384435997801861 F(6, 6, 4) = -------------, F(6, 6, 5) = 0, F(6, 6, 6) = ---------------, 72 1080 -163033753069 F(6, 6, 7) = 0, F(6, 7, 0) = 0, F(6, 7, 1) = -------------, F(6, 7, 2) = 0, 240 -119054016836171 F(6, 7, 3) = -112399321585/4, F(6, 7, 4) = 0, F(6, 7, 5) = ----------------, 360 -6227374548350707 F(6, 7, 6) = 0, F(6, 7, 7) = -----------------, F(7, 0, 0) = 0, 3780 F(7, 0, 1) = 43/6, F(7, 0, 2) = 0, F(7, 0, 3) = 26531/4, F(7, 0, 4) = 0, 49267981 92669889401 F(7, 0, 5) = --------, F(7, 0, 6) = 0, F(7, 0, 7) = -----------, 40 1680 -295559 F(7, 1, 0) = -53/6, F(7, 1, 1) = 0, F(7, 1, 2) = -------, F(7, 1, 3) = 0, 12 -118517652721 F(7, 1, 4) = -61496901/8, F(7, 1, 5) = 0, F(7, 1, 6) = -------------, 240 364249 F(7, 1, 7) = 0, F(7, 2, 0) = 0, F(7, 2, 1) = ------, F(7, 2, 2) = 0, 12 228789193 149615348961 F(7, 2, 3) = ---------, F(7, 2, 4) = 0, F(7, 2, 5) = ------------, 12 80 420088302943 F(7, 2, 6) = 0, F(7, 2, 7) = ------------, F(7, 3, 0) = -49661/4, 42 -281893463 F(7, 3, 1) = 0, F(7, 3, 2) = ----------, F(7, 3, 3) = 0, 12 -559537060963 -416482207199 F(7, 3, 4) = -------------, F(7, 3, 5) = 0, F(7, 3, 6) = -------------, 144 12 F(7, 3, 7) = 0, F(7, 4, 0) = 0, F(7, 4, 1) = 115026971/8, F(7, 4, 2) = 0, 688944765053 2499303850069 F(7, 4, 3) = ------------, F(7, 4, 4) = 0, F(7, 4, 5) = -------------, 144 36 18479063105821 -139894931 F(7, 4, 6) = 0, F(7, 4, 7) = --------------, F(7, 5, 0) = ----------, 56 40 -279274813951 F(7, 5, 1) = 0, F(7, 5, 2) = -------------, F(7, 5, 3) = 0, 80 -3050964947147 -230280114438599 F(7, 5, 4) = --------------, F(7, 5, 5) = 0, F(7, 5, 6) = ----------------, 36 360 335354989871 F(7, 5, 7) = 0, F(7, 6, 0) = 0, F(7, 6, 1) = ------------, F(7, 6, 2) = 0, 240 278791786562329 F(7, 6, 3) = 251928559147/4, F(7, 6, 4) = 0, F(7, 6, 5) = ---------------, 360 14973892230611273 -397435540711 F(7, 6, 6) = 0, F(7, 6, 7) = -----------------, F(7, 7, 0) = -------------, 3780 1680 -1122190931873 F(7, 7, 1) = 0, F(7, 7, 2) = --------------, F(7, 7, 3) = 0, 42 -97961923367113 F(7, 7, 4) = ---------------, F(7, 7, 5) = 0, 168 -17987584842404503 F(7, 7, 6) = ------------------, F(7, 7, 7) = 0 3780 -------------------------------------------- By the way, it took, 8.843, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000, 3000) = 136457363036448292966442346256144046766647758450706884\ 094948679921554937207246932260273865828976496371765442977967979879462456\ 646296149803106590390523454610935926467323519879659024765019084701864916\ 101542958657691700193139412105954047453299590988417236468398783186199073\ 611738653207632158719706709971246793686436155145078386098126504270767994\ 616246141435641950349630493783335908961869479129088452419174919956153398\ 929399049433438863560455119126260897829621232278740252055706069748292093\ 192861242477044552840236800114323115936968493460930890122359258087478467\ 168131205424439072455627212482637228069451734065948035757984575040124879\ 032865614166047264638170849826973768405150062108759760026675637823938349\ 842356225272633832745239700316097072791863375289064754146228796738852885\ 282670241247138613859752639531766991321134434744999800777775099528557624\ 423026237347185102512324018806095948732728961320043655651000419229236774\ 822653641137381529001842910915208261273636796573224017746856897365010474\ 014003151910556236320636010090005056986277704373501081961134262256903831\ 084457766012698530703917793652116551183831361142400647431380183668325522\ 655903322860808453086272589006649050602762322172867353192516451405505577\ 886784107831530866843759899593409714688552178494234098565350593232632222\ 844591275076544916621748331931761362100791306924029260328811320642893285\ 664938587003190304376181993669039973884298761093068722501246549738253401\ 415102108659966294424157671637676441768730524724275632126700711734655476\ 779316799231357042385227147763890470035948704795635211092686911861353207\ 223205531225641213119477208072260974483072705855939482617471211899721714\ 922315295285844051843956006203584174179382230427072984679183264737415682\ 286529878713324008725736989215612642273187414342206584840593849699454806\ 121670314288923316166386055439212707140052630871501206713573279049391894\ 742921282014751767468517246072108605002583803644961300102499107374757185\ 034264859619471511375969003047377623593718622364005377604047192624762392\ 108796329488243348775931627581400047611937480097701522782942407004527516\ 282872197220144761958514895411107660748802879127801698139660262761508171\ 761863557855388546671044877242304595092797744227521784787733793128006799\ 224781081409897457700628172691234579780830973120411863836663839491072799\ 300104778673077322483102740186049610843889519941202318000860894613540852\ 430316401160156648437452065103635669894601410858265472558478480858252505\ 804619944009581356915163267646187393133656814521071448765177552129817115\ 559120086835849345488140763732568693196913834590365046193871916901371881\ 094686417510720727295101829274741569034826635586667723974980499420802888\ 766133302156324149696159953690837460491373607768777024451383246734672313\ 592579052882472533321501438735255666711094465393505598131258147714733658\ 616427303860706697980126955784455413266711427146786692698404744837424828\ 991044231402994770855307580046791279944258510673156953097614787559459969\ 692090354794616810071577222572570092329445355098199163460401473833034422\ 503108197536292737048013215810296291792642150533805603006485452407278196\ 210901896970864117216273658269244757524276348295145042687608534138571705\ 062682417120095373071147288602127902971147770412739340009497314395065937\ 991815183239141400759244457748066659509889377203965164998116446857788386\ 445489831424810534117819319351652751664243377060555361169443792972425053\ 003683057675356158599776774177232370084995079765907680137593399898089021\ 382646800577074059247166881587155633279530932506425332517126157333741006\ 817102155447843555915884981055594417788012616494115332206552082360217781\ 979781445896912583033061491613331654799949030886608674112313332288628138\ 654220323339866871578636608907126915328681990118613738024005995261103569\ 880933036788155118080358150111541825243909856863122076594358222307156252\ 978572205153153574601890970385712468818365297524053583764801070729863203\ 938255754739561918099056132230121596194473996869924034446124997837559907\ 536096351165333402893624246578821803251405134396724919953169698805685065\ 579947768163158151182471731970110553998062149782620979111424260026636030\ 386110796216362293891038219699615975972543694050972578625364626927714164\ 669874824325954756305468090878699969101230908977760643905394125706524584\ 383710528164500247344461216084746911304183677034883456429246637556932518\ 604324151442244349023011670857255839377917371803186364780000586475570811\ / 2697186803201405767 / 410687650415671664589519333371491462582284404135\ / 436922825371112794699139946571105359032767036028779077847949123263682586\ 780503835288265141999645683700322352967915005371222581128234959129387931\ 417317053846479822675647831433438376322440835889407392111374603059204723\ 562112206354186743884655260285675297120430341515712683435545139099902450\ 738542069597244115081902114063064658807323036361860052157339438360313075\ 281827738998595822888851042885945579716557835546729305637590706026517192\ 663943317666657798195942322174104120888176623025432517748344838994711797\ 523167239497699423987085520407564190711106976300699757311461435081804697\ 991084031896913391298995783998515434477217468460095802875329214070078897\ 917195481783974764255686809179396207777213106869217620441958186489284005\ 609522927314823581658840012702552062807882733182010185091573589718611326\ 859268948015019227773216054837491641104433434916336084120487322074553223\ 086773896348034384065173822305582316330459452439971908768278277108868029\ 517666639597646843754383323495693172440606325118835449441868749833504709\ 562274261339740811481602173863640599381960139661774480899625170771828424\ 765355287127975469451966833836041255133576441069816444884489250913829325\ 097299522135558134712829057857011312536261654758978211079480849528106791\ 879766034910293198071174179923917093515600308357313822388167682021555602\ 700791426341054284716964069233012306443622621934746248129033278142574202\ 614123592278178686812161498582589211090399040595750563219081698150735474\ 635483119288698768370008571700794114947665557815144986338777510692434085\ 676559118264640308351996337411029219072772739022035336149457337094317039\ 898091179589191008104808573047463842033209751638619403724989547338280708\ 938192003686687003872000160481679196441554013584155020292972198922792943\ 304449945902658719373722183386147648961422809238236285320181280585520633\ 200501544972611980488976243076921358736000035050518657594782242011716039\ 109634638273448481530601824340533913094957830953102715367672618115808928\ 771391756385336872536017884590743037384057638094232139119602036637197435\ 996988949857151666357943372313752377909143391527291373314303158608565318\ 050210592393383642908749988902751984841658579221994239442896033191044634\ 984785769644113195306871890082081553753892102241828682577297511690880937\ 411347244855476139299762959663323070201653571795561119951141837889004553\ 390206503814067420609684309544797621449990940228533449070814065154796572\ 005823957886900594281691040518273160386961774031815681127122145113039568\ 949604878309491580520142738186281053625624770661369560516377316883635635\ 687534156712064469540653111601887452035854869441419337864773964127991209\ 153879218183662825941057422259665700461166881260639123643388493510921858\ 497034691282235285934851947497885678785834516654396619489963045960052926\ 144501189370426375720604095480148575485159181912438498492285154110670004\ 210898555644112177710929779983666559376735846514619494261985266586996171\ 890571262430661239286416314431827235989444659474614048643249386934349131\ 421119482915101057314207104967827389042916236393232991904239841801263378\ 497266734596881715097838690464562936567906030699374505468000939797744890\ 566773190756443503308589614537836447151819026628491062871549940745255645\ 672396298093559725697458636622729735556891352868970314247611283393346282\ 372690387630789716769697331141264855630060578381612539957481255141631359\ 380674186838018852040549133391569946751174029436169814641351196361951994\ 721530060648368118071297359105068487534646768562350193597404787505703515\ 744129905589731648899254019611948816223431506817406420232343631904175584\ 466131928591010265572979919812131208774269104410644873222599058028532957\ 597216398189298291949771276912101598386430702649165199510842262737574828\ 611193218210207025266530184799813208997811039445755463774963859817606013\ 405279515658749395660990856890954680581278018649204740692765613295828588\ 017676628024378041587661840948793980832826320384024854024822223440668208\ 978698751390063786207592852701031346612036112988895663346836891756434052\ 698641385147234897774948870004373597847361519275035078606587269917853361\ 504500458148170811711141110595636959005002024288060953213005692025926564\ 284233811491194967144548095093140096004861375081250421095832861209704049\ 853546353257817324637794938026251527327247077549705260360526106816064419\ 249413232034883159081524360164298486718717420817014821595694654772226414\ 951673051966297942927338403784106635554269040184269995966633921311019457\ 735369901155518224204277547585418598585642836177853160572213216465462151\ 006788433935987002209966421344701425301105505842228744402355223586479944\ 499919292723469950752498452571868241917103891191968915493099035637716245\ 480188122499049086449323706611689128300546262105089365031364069782838711\ 996763900345185693022853719206531929803093454523157896280612502173147557\ 752967430434108316949240276975472546328187128997352432113249839478620232\ 883790849378655308650989510187325642422225759949139467940455078671301533\ 320795541340200087029782209663148805375768176394952000880267071464145071\ 079422210583870809331273979033348014985146420575453939414419519149172319\ 371529779125704344144429509307892455111554863793247804525420906464392489\ 477720213520109897298657868767322419233676970062238837159087211733648059\ 807674438655042786292905036718203702274520974418590255335026611017574418\ 269477946858902234433803833929907747646608549548537941829119028210347835\ 058686872683863008547560402811275360341727065221837262073466363304322730\ 001619060068959471677819306150177531011105130623205834326242081265566123\ 034150673705236466771721444488983039964495077458108930357536286056249943\ 306696550898392183592819809135078889081772148681742256126678993388867951\ 295679177519353728503127310760998182934770676865634409314732399868362103\ 856804218668173724018426282276970892779625912975608032768564370563932708\ 125953283880736425446165994248145453512909585033770934276130049218875685\ 739242789297818090294178432356776719206302625761783429860474265599486861\ 223410539216667972868226266655565634741535379823850396709959269887441011\ 345102402015359278671187205083273150579715365798111464647507900612306320\ 888009738649681893513946046639091709622772796599060604816852072812515598\ 104835687507657214448993041673241200991988344124327356426707676784382934\ 740057513448823695941713990162521507795639835386917491891910482063090085\ 775222601628801128761844587573842488350070808124423792856071544782200900\ 270732030241810008654417322275524247884190087793679086002449594311033947\ 775742116125441655170745458597532753735817566984114861353792690033992538\ 102148046059440065711701466576653059464836966059831464284929619370278948\ 415254317491461192648985620870841320480040134188135814879774421758227133\ 527983530129571751143159835675772811504567616219039429267806747044527286\ 716734569100082332908197074894175709978310649340984343921421829701687768\ 479449287968241322579880206594112350470344250677272649792592783543339318\ 333518035267150343227673490078725062898645405079740819400637802498033712\ 775760789121712883149989552104403077154542884992807977944816217162647763\ 101601855139565835111620346855373886263374356496653819301767830911285244\ 203802727232369483430894656742440129251386346200340637202182734751970157\ 997117641747803597235397869225963919016370618963731850992742046590197824\ 189665768511431408448104712855704656441340923975275383720941530733598878\ 591038860167721502258410476378606176473295208867900506841695185494686699\ 763212861861259948218997184002404104573823792230217484656276851707050382\ 564639948721780840936360377867543229255759490707435673512218267312109519\ 482269281235763200000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000 This took, 36.527, seconds. --------------------------- A Pure Recurrence Scheme that enables Linear-Time and Constant-Space Calcula\ tion of the Taylor coefficients of the function with, 3, variables , x[1], x[2], x[3] 2 2 2 exp(x[1] + 92 x[1] x[2] + 81 x[1] x[3] + x[2] - 86 x[2] x[3] + x[3] ) By Shalosh B. Ekhad Theorem: Let , F(m[1], m[2], m[3]), be the coefficient of, m[1] m[2] m[3] x[1] x[2] x[3] , in the Taylor expansion (around the origing)\ of the multivariable function (that may also be viewed as a formal powe\ r series) of, x[1], x[2], x[3] 2 2 2 exp(x[1] + 92 x[1] x[2] + 81 x[1] x[3] + x[2] - 86 x[2] x[3] + x[3] ) The following PURE recurrence relations hold in each of the, 3, discrete variables, m[1], m[2], m[3] 2 F(m[1], m[2], m[3]) = -1/336 (62074294165426646800125 m[1] - 497640452648014046194525 m[1] m[2] + 497805956484997025431935 m[1] m[3] 2 + 51878682771387064356400 m[2] - 127245308492993245333772 m[2] m[3] 2 + 75228988827617004941316 m[3] - 373817281924473429378240 m[1] + 1492155736492530045303153 m[2] - 1495842526729988893817379 m[3] + 499160810371231333567755) F(-2 + m[1], m[2], m[3])/(m[1] (-1 + m[1]) %1) 3 - 1/672 (211898878449734988077848425 m[1] 2 - 1650873985845726740124543400 m[1] m[2] 2 + 1686328957685849459261611914 m[1] m[3] 2 - 449716148272724100086420400 m[1] m[2] + 538169526828946592771505592 m[1] m[2] m[3] 2 - 90528572764688721815646819 m[1] m[3] 3 + 1961652406898438706452000000 m[2] 2 - 5850907133766209440805880000 m[2] m[3] 2 + 5817054059045756121967772400 m[2] m[3] 3 2 - 1927798952210180515470858084 m[3] - 2106225639171222280736744736 m[1] + 13247958184191957311256122892 m[1] m[2] - 13465935137094491715718253715 m[1] m[3] 2 + 1236521162477833044531716400 m[2] - 1212527979889078979941257272 m[2] m[3] 2 - 16741300942179489279034773 m[3] + 6701019173172302189113897479 m[1] - 26579211890429077732628812372 m[2] + 26882790024566342942365405437 m[3] - 6665982998872928939925092424) F(m[1] - 4, m[2], m[3])/((-3 + m[1]) m[1] 2 (-1 + m[1]) (-2 + m[1]) %1) + 1/336 (21296957298414737875954954650 m[1] - 189331660001732892282966899000 m[1] m[2] + 146657152967253652794347150742 m[1] m[3] 2 + 164924523454041564419298601600 m[2] + 1000024526824124628243069632 m[2] m[3] 2 - 164015791964994626374835080152 m[3] - 170914897075387016783155039533 m[1] + 1050210979631273122953384196432 m[2] - 959050696198743419667353717652 m[3] + 260544949680305242827726191325) F(m[1] - 6, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) - 278743282815 ------------ (25214132327637825 m[1] - 51596052752694339 56 - 199474320187433200 m[2] + 198320171110763436 m[3]) F(m[1] - 8, m[2], m[3])/((-3 + m[1]) m[1] (-1 + m[1]) (-2 + m[1]) %1) %1 := 25214132327637825 m[1] - 102024317407969989 - 199474320187433200 m[2] + 198320171110763436 m[3] 2 F(m[1], m[2], m[3]) = 2/6557 (582323087808009769646425 m[1] - 13514494856817988161764675 m[1] m[2] 2 - 1861904715911872781702679 m[1] m[3] + 906500431679019740043000 m[2] 2 + 13517770551979582324385290 m[2] m[3] + 1277311358674310355105542 m[3] + 40530304892066550364870421 m[1] - 5459702474336297033063435 m[2] - 40589634873015590334201993 m[3] + 7290408755703535446371740) F(m[1], -2 + m[2], m[3])/(m[2] (-1 + m[2]) %1) + 4/6557 ( 3 14395262994779621813045687500 m[1] 2 - 2448950859447098424205094450 m[1] m[2] 2 - 43045556626068041759328615000 m[1] m[3] 2 - 12129452868279126535176489325 m[1] m[2] + 2116075704463367304270952966 m[1] m[2] m[3] 2 + 42905779628571414612478621200 m[1] m[3] 3 + 822662467483684578827827400 m[2] 2 + 12276133001298224090047751902 m[2] m[3] 2 + 323867866352637709172147428 m[2] m[3] 3 2 - 14255485504402375590592977952 m[3] + 7260453324236016588297403075 m[1] + 97207026124342208277283999116 m[1] m[2] - 4793028909988022676901502981 m[1] m[3] 2 - 8173556428343857240310024173 m[2] - 98109886235499867321062239650 m[2] m[3] 2 - 2435912726199765206660389454 m[3] - 194751792709480686935382162156 m[1] + 25993040004045638015770411422 m[2] + 196011829606648786735307838471 m[3] - 25845610653044465217265139427) F(m[1], m[2] - 4, m[3])/((-3 + m[2]) m[2] 264 2 (-1 + m[2]) (-2 + m[2]) %1) - ---- (41400773966483482539767753975 m[1] 6557 - 44973079945088967098637299375 m[1] m[2] + 140149312965299830515788927 m[1] m[3] 2 + 2838035635126151568055789400 m[2] + 39265547467700895071108408162 m[2] m[3] 2 - 41272032568212565681041213622 m[3] + 258092038938649575363813678602 m[1] - 22789949770389760550497796863 m[2] - 245859700834509949724023595897 m[3] + 34776191461246989564366702675) F(m[1], m[2] - 6, m[3])/((-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1) - 41479653156480 -------------- (532294339702400275 m[1] - 35808055156585400 m[2] 6557 + 73362712250489283 - 530565880125622642 m[3]) F(m[1], m[2] - 8, m[3])/( (-3 + m[2]) m[2] (-1 + m[2]) (-2 + m[2]) %1) %1 := 532294339702400275 m[1] - 35808055156585400 m[2] + 144978822563660083 - 530565880125622642 m[3] 2 F(m[1], m[2], m[3]) = -1/4230 (53801522974050278109988734 m[1] - 78634340065114614212215420 m[1] m[2] 2 + 643956988515749118130195566 m[1] m[3] + 24758493518223963120929696 m[2] 2 - 643848447805562305734724376 m[2] m[3] + 38406724667656435596755340 m[3] - 1933312974932707118115683463 m[1] + 1931162441562879801081105228 m[2] - 231226236858574119860896320 m[3] + 308734151358110543206679475) F(m[1], m[2], -2 + m[3])/(m[3] (-1 + m[3]) %1) + 1/8460 ( 3 2376366972972496816472115229989 m[1] 2 - 7147305365035638641984664814152 m[1] m[2] 2 - 48090583256626156957859261916 m[1] m[3] 2 + 7165556296939615615064103448512 m[1] m[2] - 310060621612729091419909498520 m[1] m[2] m[3] 2 - 2081495314183706878635270590355 m[1] m[3] 3 - 2394617944444248791597165684224 m[2] 2 + 359188873203072221098974021696 m[2] m[3] 2 + 2062444850710067988086539674200 m[2] m[3] 3 - 124084283301055439158245998850 m[3] 2 + 363532885946475333038096407866 m[1] + 693783503516348996254395274900 m[1] m[2] + 16640444528291184815979051791823 m[1] m[3] 2 - 1060946955712685531479919038176 m[2] - 16523507880028716975375004832088 m[2] m[3] 2 + 1234031984081744020761307395735 m[3] - 33255739122344819730595885648203 m[1] + 33093320545347197783852481321788 m[2] - 3928239060979144483845127844505 m[3] + 3909831971231558473811427488400) F(m[1], m[2], m[3] - 4)/((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) - 11 2 ---- (16032404210179736969913930527349 m[1] 4230 - 43177350596954178822220128125 m[1] m[2] - 15465807984707043981236675188623 m[1] m[3] 2 - 16071099923657989498336923455464 m[2] + 17429647426396632304426355805458 m[2] m[3] 2 - 984655660265906312353801728870 m[3] + 96220565085213753068903615992890 m[1] - 100398046424555441407865426643025 m[2] + 7896048995633414966522592125445 m[3] - 12021431372853413531343496298955) F(m[1], m[2], m[3] - 6)/((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) - 2679098559368 ------------- (1332161306048987190 m[3] - 2721839244324620895 705 - 22280941902440220376 m[2] + 22224191534908870221 m[1]) F(m[1], m[2], m[3] - 8)/((-3 + m[3]) m[3] (-1 + m[3]) (-2 + m[3]) %1) %1 := 22224191534908870221 m[1] - 22280941902440220376 m[2] + 1332161306048987190 m[3] - 5386161856422595275 and in Maple notation F(m[1],m[2],m[3]) = -1/336*(62074294165426646800125*m[1]^2-\ 497640452648014046194525*m[1]*m[2]+497805956484997025431935*m[1]*m[3]+ 51878682771387064356400*m[2]^2-127245308492993245333772*m[2]*m[3]+ 75228988827617004941316*m[3]^2-373817281924473429378240*m[1]+ 1492155736492530045303153*m[2]-1495842526729988893817379*m[3]+ 499160810371231333567755)/m[1]/(-1+m[1])/(25214132327637825*m[1]-\ 102024317407969989-199474320187433200*m[2]+198320171110763436*m[3])*F(-2+m[1],m [2],m[3])-1/672*(211898878449734988077848425*m[1]^3-\ 1650873985845726740124543400*m[1]^2*m[2]+1686328957685849459261611914*m[1]^2*m[ 3]-449716148272724100086420400*m[1]*m[2]^2+538169526828946592771505592*m[1]*m[2 ]*m[3]-90528572764688721815646819*m[1]*m[3]^2+1961652406898438706452000000*m[2] ^3-5850907133766209440805880000*m[2]^2*m[3]+5817054059045756121967772400*m[2]*m [3]^2-1927798952210180515470858084*m[3]^3-2106225639171222280736744736*m[1]^2+ 13247958184191957311256122892*m[1]*m[2]-13465935137094491715718253715*m[1]*m[3] +1236521162477833044531716400*m[2]^2-1212527979889078979941257272*m[2]*m[3]-\ 16741300942179489279034773*m[3]^2+6701019173172302189113897479*m[1]-\ 26579211890429077732628812372*m[2]+26882790024566342942365405437*m[3]-\ 6665982998872928939925092424)/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[1])/( 25214132327637825*m[1]-102024317407969989-199474320187433200*m[2]+ 198320171110763436*m[3])*F(m[1]-4,m[2],m[3])+1/336*( 21296957298414737875954954650*m[1]^2-189331660001732892282966899000*m[1]*m[2]+ 146657152967253652794347150742*m[1]*m[3]+164924523454041564419298601600*m[2]^2+ 1000024526824124628243069632*m[2]*m[3]-164015791964994626374835080152*m[3]^2-\ 170914897075387016783155039533*m[1]+1050210979631273122953384196432*m[2]-\ 959050696198743419667353717652*m[3]+260544949680305242827726191325)/(-3+m[1])/m [1]/(-1+m[1])/(-2+m[1])/(25214132327637825*m[1]-102024317407969989-\ 199474320187433200*m[2]+198320171110763436*m[3])*F(m[1]-6,m[2],m[3])-\ 278743282815/56*(25214132327637825*m[1]-51596052752694339-199474320187433200*m[ 2]+198320171110763436*m[3])/(-3+m[1])/m[1]/(-1+m[1])/(-2+m[1])/( 25214132327637825*m[1]-102024317407969989-199474320187433200*m[2]+ 198320171110763436*m[3])*F(m[1]-8,m[2],m[3]) F(m[1],m[2],m[3]) = 2/6557*(582323087808009769646425*m[1]^2-\ 13514494856817988161764675*m[1]*m[2]-1861904715911872781702679*m[1]*m[3]+ 906500431679019740043000*m[2]^2+13517770551979582324385290*m[2]*m[3]+ 1277311358674310355105542*m[3]^2+40530304892066550364870421*m[1]-\ 5459702474336297033063435*m[2]-40589634873015590334201993*m[3]+ 7290408755703535446371740)/m[2]/(-1+m[2])/(532294339702400275*m[1]-\ 35808055156585400*m[2]+144978822563660083-530565880125622642*m[3])*F(m[1],-2+m[ 2],m[3])+4/6557*(14395262994779621813045687500*m[1]^3-\ 2448950859447098424205094450*m[1]^2*m[2]-43045556626068041759328615000*m[1]^2*m [3]-12129452868279126535176489325*m[1]*m[2]^2+2116075704463367304270952966*m[1] *m[2]*m[3]+42905779628571414612478621200*m[1]*m[3]^2+ 822662467483684578827827400*m[2]^3+12276133001298224090047751902*m[2]^2*m[3]+ 323867866352637709172147428*m[2]*m[3]^2-14255485504402375590592977952*m[3]^3+ 7260453324236016588297403075*m[1]^2+97207026124342208277283999116*m[1]*m[2]-\ 4793028909988022676901502981*m[1]*m[3]-8173556428343857240310024173*m[2]^2-\ 98109886235499867321062239650*m[2]*m[3]-2435912726199765206660389454*m[3]^2-\ 194751792709480686935382162156*m[1]+25993040004045638015770411422*m[2]+ 196011829606648786735307838471*m[3]-25845610653044465217265139427)/(-3+m[2])/m[ 2]/(-1+m[2])/(-2+m[2])/(532294339702400275*m[1]-35808055156585400*m[2]+ 144978822563660083-530565880125622642*m[3])*F(m[1],m[2]-4,m[3])-264/6557*( 41400773966483482539767753975*m[1]^2-44973079945088967098637299375*m[1]*m[2]+ 140149312965299830515788927*m[1]*m[3]+2838035635126151568055789400*m[2]^2+ 39265547467700895071108408162*m[2]*m[3]-41272032568212565681041213622*m[3]^2+ 258092038938649575363813678602*m[1]-22789949770389760550497796863*m[2]-\ 245859700834509949724023595897*m[3]+34776191461246989564366702675)/(-3+m[2])/m[ 2]/(-1+m[2])/(-2+m[2])/(532294339702400275*m[1]-35808055156585400*m[2]+ 144978822563660083-530565880125622642*m[3])*F(m[1],m[2]-6,m[3])-41479653156480/ 6557*(532294339702400275*m[1]-35808055156585400*m[2]+73362712250489283-\ 530565880125622642*m[3])/(-3+m[2])/m[2]/(-1+m[2])/(-2+m[2])/(532294339702400275 *m[1]-35808055156585400*m[2]+144978822563660083-530565880125622642*m[3])*F(m[1] ,m[2]-8,m[3]) F(m[1],m[2],m[3]) = -1/4230*(53801522974050278109988734*m[1]^2-\ 78634340065114614212215420*m[1]*m[2]+643956988515749118130195566*m[1]*m[3]+ 24758493518223963120929696*m[2]^2-643848447805562305734724376*m[2]*m[3]+ 38406724667656435596755340*m[3]^2-1933312974932707118115683463*m[1]+ 1931162441562879801081105228*m[2]-231226236858574119860896320*m[3]+ 308734151358110543206679475)/m[3]/(-1+m[3])/(22224191534908870221*m[1]-\ 22280941902440220376*m[2]+1332161306048987190*m[3]-5386161856422595275)*F(m[1], m[2],-2+m[3])+1/8460*(2376366972972496816472115229989*m[1]^3-\ 7147305365035638641984664814152*m[1]^2*m[2]-48090583256626156957859261916*m[1]^ 2*m[3]+7165556296939615615064103448512*m[1]*m[2]^2-\ 310060621612729091419909498520*m[1]*m[2]*m[3]-2081495314183706878635270590355*m [1]*m[3]^2-2394617944444248791597165684224*m[2]^3+ 359188873203072221098974021696*m[2]^2*m[3]+2062444850710067988086539674200*m[2] *m[3]^2-124084283301055439158245998850*m[3]^3+363532885946475333038096407866*m[ 1]^2+693783503516348996254395274900*m[1]*m[2]+16640444528291184815979051791823* m[1]*m[3]-1060946955712685531479919038176*m[2]^2-\ 16523507880028716975375004832088*m[2]*m[3]+1234031984081744020761307395735*m[3] ^2-33255739122344819730595885648203*m[1]+33093320545347197783852481321788*m[2]-\ 3928239060979144483845127844505*m[3]+3909831971231558473811427488400)/(-3+m[3]) /m[3]/(-1+m[3])/(-2+m[3])/(22224191534908870221*m[1]-22280941902440220376*m[2]+ 1332161306048987190*m[3]-5386161856422595275)*F(m[1],m[2],m[3]-4)-11/4230*( 16032404210179736969913930527349*m[1]^2-43177350596954178822220128125*m[1]*m[2] -15465807984707043981236675188623*m[1]*m[3]-16071099923657989498336923455464*m[ 2]^2+17429647426396632304426355805458*m[2]*m[3]-984655660265906312353801728870* m[3]^2+96220565085213753068903615992890*m[1]-100398046424555441407865426643025* m[2]+7896048995633414966522592125445*m[3]-12021431372853413531343496298955)/(-3 +m[3])/m[3]/(-1+m[3])/(-2+m[3])/(22224191534908870221*m[1]-22280941902440220376 *m[2]+1332161306048987190*m[3]-5386161856422595275)*F(m[1],m[2],m[3]-6)-\ 2679098559368/705*(1332161306048987190*m[3]-2721839244324620895-\ 22280941902440220376*m[2]+22224191534908870221*m[1])/(-3+m[3])/m[3]/(-1+m[3])/( -2+m[3])/(22224191534908870221*m[1]-22280941902440220376*m[2]+ 1332161306048987190*m[3]-5386161856422595275)*F(m[1],m[2],m[3]-8) Subject to the initial conditions F(0, 0, 0) = 1, F(0, 0, 1) = 0, F(0, 0, 2) = 1, F(0, 0, 3) = 0, F(0, 0, 4) = 1/2, F(0, 0, 5) = 0, F(0, 0, 6) = 1/6, F(0, 0, 7) = 0, F(0, 1, 0) = 0, F(0, 1, 1) = -86, F(0, 1, 2) = 0, F(0, 1, 3) = -86, F(0, 1, 4) = 0, F(0, 1, 5) = -43, F(0, 1, 6) = 0, F(0, 1, 7) = -43/3, F(0, 2, 0) = 1, F(0, 2, 1) = 0, F(0, 2, 2) = 3699, F(0, 2, 3) = 0, F(0, 2, 4) = 7397/2, F(0, 2, 5) = 0, F(0, 2, 6) = 11095/6, F(0, 2, 7) = 0, F(0, 3, 0) = 0, F(0, 3, 1) = -86, F(0, 3, 2) = 0, F(0, 3, 3) = -318286/3, F(0, 3, 4) = 0, F(0, 3, 5) = -318157/3, F(0, 3, 6) = 0, F(0, 3, 7) = -53019, F(0, 4, 0) = 1/2, F(0, 4, 1) = 0, F(0, 4, 2) = 7397/2, F(0, 4, 3) = 0, 27394787 F(0, 4, 4) = --------, F(0, 4, 5) = 0, F(0, 4, 6) = 9124199/4, 12 F(0, 4, 7) = 0, F(0, 5, 0) = 0, F(0, 5, 1) = -43, F(0, 5, 2) = 0, -393082823 F(0, 5, 3) = -318157/3, F(0, 5, 4) = 0, F(0, 5, 5) = ----------, 10 -392552633 F(0, 5, 6) = 0, F(0, 5, 7) = ----------, F(0, 6, 0) = 1/6, F(0, 6, 1) = 0, 10 F(0, 6, 2) = 11095/6, F(0, 6, 3) = 0, F(0, 6, 4) = 9124199/4, 33850743773 F(0, 6, 5) = 0, F(0, 6, 6) = -----------, F(0, 6, 7) = 0, F(0, 7, 0) = 0, 60 F(0, 7, 1) = -43/3, F(0, 7, 2) = 0, F(0, 7, 3) = -53019, F(0, 7, 4) = 0, -392552633 -1457937298037 F(0, 7, 5) = ----------, F(0, 7, 6) = 0, F(0, 7, 7) = --------------, 10 210 F(1, 0, 0) = 0, F(1, 0, 1) = 81, F(1, 0, 2) = 0, F(1, 0, 3) = 81, F(1, 0, 4) = 0, F(1, 0, 5) = 81/2, F(1, 0, 6) = 0, F(1, 0, 7) = 27/2, F(1, 1, 0) = 92, F(1, 1, 1) = 0, F(1, 1, 2) = -6874, F(1, 1, 3) = 0, F(1, 1, 4) = -6920, F(1, 1, 5) = 0, F(1, 1, 6) = -10403/3, F(1, 1, 7) = 0, F(1, 2, 0) = 0, F(1, 2, 1) = -7831, F(1, 2, 2) = 0, F(1, 2, 3) = 291707, F(1, 2, 4) = 0, F(1, 2, 5) = 591245/2, F(1, 2, 6) = 0, F(1, 2, 7) = 890783/6, F(1, 3, 0) = 92, F(1, 3, 1) = 0, F(1, 3, 2) = 333342, F(1, 3, 3) = 0, F(1, 3, 4) = -8253460, F(1, 3, 5) = 0, F(1, 3, 6) = -25260347/3, F(1, 3, 7) = 0, F(1, 4, 0) = 0, F(1, 4, 1) = -15743/2, F(1, 4, 2) = 0, F(1, 4, 3) = -56767153/6, 2101895971 F(1, 4, 4) = 0, F(1, 4, 5) = ----------, F(1, 4, 6) = 0, 12 F(1, 4, 7) = 719549127/4, F(1, 5, 0) = 46, F(1, 5, 1) = 0, F(1, 5, 2) = 336779, F(1, 5, 3) = 0, F(1, 5, 4) = 604309384/3, -29741142893 F(1, 5, 5) = 0, F(1, 5, 6) = ------------, F(1, 5, 7) = 0, F(1, 6, 0) = 0, 10 F(1, 6, 1) = -7885/2, F(1, 6, 2) = 0, F(1, 6, 3) = -57642193/6, -68631938837 F(1, 6, 4) = 0, F(1, 6, 5) = ------------, F(1, 6, 6) = 0, 20 841740397399 F(1, 6, 7) = ------------, F(1, 7, 0) = 46/3, F(1, 7, 1) = 0, 20 F(1, 7, 2) = 506887/3, F(1, 7, 3) = 0, F(1, 7, 4) = 205562038, 1461743923739 F(1, 7, 5) = 0, F(1, 7, 6) = -------------, F(1, 7, 7) = 0, F(2, 0, 0) = 1, 30 F(2, 0, 1) = 0, F(2, 0, 2) = 6563/2, F(2, 0, 3) = 0, F(2, 0, 4) = 3281, 19685 F(2, 0, 5) = 0, F(2, 0, 6) = -----, F(2, 0, 7) = 0, F(2, 1, 0) = 0, 12 F(2, 1, 1) = 7366, F(2, 1, 2) = 0, F(2, 1, 3) = -274757, F(2, 1, 4) = 0, F(2, 1, 5) = -278440, F(2, 1, 6) = 0, F(2, 1, 7) = -839003/6, F(2, 2, 0) = 4233, F(2, 2, 1) = 0, F(2, 2, 2) = -1259321/2, F(2, 2, 3) = 0, F(2, 2, 4) = 11499512, F(2, 2, 5) = 0, F(2, 2, 6) = 47260191/4, F(2, 2, 7) = 0, F(2, 3, 0) = 0, F(2, 3, 1) = -356586, F(2, 3, 2) = 0, F(2, 3, 3) = 80438333/3, F(2, 3, 4) = 0, F(2, 3, 5) = -962317642/3, F(2, 3, 6) = 0, F(2, 3, 7) = -2005430203/6, F(2, 4, 0) = 8465/2, F(2, 4, 1) = 0, F(2, 4, 2) = 60074539/4, F(2, 4, 3) = 0, 161012511149 F(2, 4, 4) = -4563327263/6, F(2, 4, 5) = 0, F(2, 4, 6) = ------------, 24 F(2, 4, 7) = 0, F(2, 5, 0) = 0, F(2, 5, 1) = -360269, F(2, 5, 2) = 0, 242642044358 F(2, 5, 3) = -843362685/2, F(2, 5, 4) = 0, F(2, 5, 5) = ------------, 15 -2244321380389 F(2, 5, 6) = 0, F(2, 5, 7) = --------------, F(2, 6, 0) = 12697/6, 20 184008143 F(2, 6, 1) = 0, F(2, 6, 2) = ---------, F(2, 6, 3) = 0, 12 -33026970526781 F(2, 6, 4) = 26637820405/3, F(2, 6, 5) = 0, F(2, 6, 6) = ---------------, 120 F(2, 6, 7) = 0, F(2, 7, 0) = 0, F(2, 7, 1) = -542245/3, F(2, 7, 2) = 0, F(2, 7, 3) = -2610801145/6, F(2, 7, 4) = 0, F(2, 7, 5) = -747837260247/5, 1639014154051373 F(2, 7, 6) = 0, F(2, 7, 7) = ----------------, F(3, 0, 0) = 0, 420 F(3, 0, 1) = 81, F(3, 0, 2) = 0, F(3, 0, 3) = 177309/2, F(3, 0, 4) = 0, F(3, 0, 5) = 88614, F(3, 0, 6) = 0, F(3, 0, 7) = 177201/4, F(3, 1, 0) = 92, F(3, 1, 1) = 0, F(3, 1, 2) = 294932, F(3, 1, 3) = 0, F(3, 1, 4) = -7322435, F(3, 1, 5) = 0, F(3, 1, 6) = -22409657/3, F(3, 1, 7) = 0, F(3, 2, 0) = 0, F(3, 2, 1) = 334961, F(3, 2, 2) = 0, F(3, 2, 3) = -50464487/2, F(3, 2, 4) = 0, F(3, 2, 5) = 302145079, F(3, 2, 6) = 0, 3777804331 F(3, 2, 7) = ----------, F(3, 3, 0) = 389620/3, F(3, 3, 1) = 0, 12 F(3, 3, 2) = -86145548/3, F(3, 3, 3) = 0, F(3, 3, 4) = 3213283175/3, F(3, 3, 5) = 0, F(3, 3, 6) = -74737296517/9, F(3, 3, 7) = 0, F(3, 4, 0) = 0, 14657450303 F(3, 4, 1) = -64957645/6, F(3, 4, 2) = 0, F(3, 4, 3) = -----------, 12 F(3, 4, 4) = 0, F(3, 4, 5) = -30255225470, F(3, 4, 6) = 0, 12312649677889 F(3, 4, 7) = --------------, F(3, 5, 0) = 389482/3, F(3, 5, 1) = 0, 72 F(3, 5, 2) = 451068722, F(3, 5, 3) = 0, F(3, 5, 4) = -207448667705/6, 57646003110773 F(3, 5, 5) = 0, F(3, 5, 6) = --------------, F(3, 5, 7) = 0, F(3, 6, 0) = 0, 90 -450862111309 F(3, 6, 1) = -65962447/6, F(3, 6, 2) = 0, F(3, 6, 3) = -------------, 36 66016874873411 F(3, 6, 4) = 0, F(3, 6, 5) = --------------, F(3, 6, 6) = 0, 90 -433772125541689 F(3, 6, 7) = ----------------, F(3, 7, 0) = 64906, F(3, 7, 1) = 0, 40 4692350806231 F(3, 7, 2) = 1396426406/3, F(3, 7, 3) = 0, F(3, 7, 4) = -------------, 18 -373381914249509 F(3, 7, 5) = 0, F(3, 7, 6) = ----------------, F(3, 7, 7) = 0, 30 F(4, 0, 0) = 1/2, F(4, 0, 1) = 0, F(4, 0, 2) = 3281, F(4, 0, 3) = 0, 43086089 F(4, 0, 4) = 14375153/8, F(4, 0, 5) = 0, F(4, 0, 6) = --------, 24 F(4, 0, 7) = 0, F(4, 1, 0) = 0, F(4, 1, 1) = 7409, F(4, 1, 2) = 0, F(4, 1, 3) = 7874048, F(4, 1, 4) = 0, F(4, 1, 5) = -585521627/4, -1803794351 F(4, 1, 6) = 0, F(4, 1, 7) = -----------, F(4, 2, 0) = 8465/2, 12 F(4, 2, 1) = 0, F(4, 2, 2) = 13251566, F(4, 2, 3) = 0, 142860912079 F(4, 2, 4) = -5388953439/8, F(4, 2, 5) = 0, F(4, 2, 6) = ------------, 24 F(4, 2, 7) = 0, F(4, 3, 0) = 0, F(4, 3, 1) = 10155745, F(4, 3, 2) = 0, 341743343695 F(4, 3, 3) = -3445252982/3, F(4, 3, 4) = 0, F(4, 3, 5) = ------------, 12 -1933022936549 35870435 F(4, 3, 6) = 0, F(4, 3, 7) = --------------, F(4, 4, 0) = --------, 12 12 F(4, 4, 1) = 0, F(4, 4, 2) = -5233031615/6, F(4, 4, 3) = 0, 2351488373891 -115244525055329 F(4, 4, 4) = -------------, F(4, 4, 5) = 0, F(4, 4, 6) = ----------------, 48 144 F(4, 4, 7) = 0, F(4, 5, 0) = 0, F(4, 5, 1) = -1479332621/6, F(4, 5, 2) = 0, -166550564540293 F(4, 5, 3) = 111050242090/3, F(4, 5, 4) = 0, F(4, 5, 5) = ----------------, 120 6065082072506347 F(4, 5, 6) = 0, F(4, 5, 7) = ----------------, F(4, 6, 0) = 11948347/4, 360 F(4, 6, 1) = 0, F(4, 6, 2) = 30478873060/3, F(4, 6, 3) = 0, -16710243099941 F(4, 6, 4) = ---------------, F(4, 6, 5) = 0, 16 21202517390294471 F(4, 6, 6) = -----------------, F(4, 6, 7) = 0, F(4, 7, 0) = 0, 720 F(4, 7, 1) = -503264149/2, F(4, 7, 2) = 0, F(4, 7, 3) = -2509582336721/9, 1589406484701695 F(4, 7, 4) = 0, F(4, 7, 5) = ----------------, F(4, 7, 6) = 0, 72 -419580467862056017 F(4, 7, 7) = -------------------, F(5, 0, 0) = 0, F(5, 0, 1) = 81/2, 840 1165805217 F(5, 0, 2) = 0, F(5, 0, 3) = 88614, F(5, 0, 4) = 0, F(5, 0, 5) = ----------, 40 1164033207 F(5, 0, 6) = 0, F(5, 0, 7) = ----------, F(5, 1, 0) = 46, F(5, 1, 1) = 0, 40 F(5, 1, 2) = 298369, F(5, 1, 3) = 0, F(5, 1, 4) = 315386911/2, -140479072523 F(5, 1, 5) = 0, F(5, 1, 6) = -------------, F(5, 1, 7) = 0, F(5, 2, 0) = 0, 60 F(5, 2, 1) = 677753/2, F(5, 2, 2) = 0, F(5, 2, 3) = 349464955, -539406886663 F(5, 2, 4) = 0, F(5, 2, 5) = -------------, F(5, 2, 6) = 0, 40 11254946935139 F(5, 2, 7) = --------------, F(5, 3, 0) = 389482/3, F(5, 3, 1) = 0, 120 F(5, 3, 2) = 1190597431/3, F(5, 3, 3) = 0, F(5, 3, 4) = -183423127535/6, 102160540012651 F(5, 3, 5) = 0, F(5, 3, 6) = ---------------, F(5, 3, 7) = 0, 180 2771523427 F(5, 4, 0) = 0, F(5, 4, 1) = ----------, F(5, 4, 2) = 0, 12 313327607119811 F(5, 4, 3) = -34771488640, F(5, 4, 4) = 0, F(5, 4, 5) = ---------------, 240 -11419613252720189 F(5, 4, 6) = 0, F(5, 4, 7) = ------------------, F(5, 5, 0) = 275266323/5, 720 -595852026217 F(5, 5, 1) = 0, F(5, 5, 2) = -------------, F(5, 5, 3) = 0, 30 89234716365833 -1478152442691793 F(5, 5, 4) = --------------, F(5, 5, 5) = 0, F(5, 5, 6) = -----------------, 60 40 -89852534577 F(5, 5, 7) = 0, F(5, 6, 0) = 0, F(5, 6, 1) = ------------, F(5, 6, 2) = 0, 20 75661603829531 F(5, 6, 3) = --------------, F(5, 6, 4) = 0, 90 -30357695545510421 F(5, 6, 5) = ------------------, F(5, 6, 6) = 0, 720 563552642687103839 F(5, 6, 7) = ------------------, F(5, 7, 0) = 274941793/5, F(5, 7, 1) = 0, 720 1830468386819 -4253355595560467 F(5, 7, 2) = -------------, F(5, 7, 3) = 0, F(5, 7, 4) = -----------------, 10 180 64402570412629757 F(5, 7, 5) = 0, F(5, 7, 6) = -----------------, F(5, 7, 7) = 0, 72 19685 F(6, 0, 0) = 1/6, F(6, 0, 1) = 0, F(6, 0, 2) = -----, F(6, 0, 3) = 0, 12 43086089 283721528621 F(6, 0, 4) = --------, F(6, 0, 5) = 0, F(6, 0, 6) = ------------, 24 720 F(6, 0, 7) = 0, F(6, 1, 0) = 0, F(6, 1, 1) = 11135/3, F(6, 1, 2) = 0, 151617610811 F(6, 1, 3) = 48068473/6, F(6, 1, 4) = 0, F(6, 1, 5) = ------------, 60 -11236206235307 F(6, 1, 6) = 0, F(6, 1, 7) = ---------------, F(6, 2, 0) = 12697/6, 360 F(6, 2, 1) = 0, F(6, 2, 2) = 54268051/4, F(6, 2, 3) = 0, 165868883605 -155495413682551 F(6, 2, 4) = ------------, F(6, 2, 5) = 0, F(6, 2, 6) = ----------------, 24 720 F(6, 2, 7) = 0, F(6, 3, 0) = 0, F(6, 3, 1) = 31002071/3, F(6, 3, 2) = 0, 186000213527 -109796725112947 F(6, 3, 3) = ------------, F(6, 3, 4) = 0, F(6, 3, 5) = ----------------, 18 180 1085577303330827 F(6, 3, 6) = 0, F(6, 3, 7) = ----------------, F(6, 4, 0) = 11948347/4, 120 213900375245 F(6, 4, 1) = 0, F(6, 4, 2) = ------------, F(6, 4, 3) = 0, 24 -132743996475149 F(6, 4, 4) = ----------------, F(6, 4, 5) = 0, 144 37527080668072117 F(6, 4, 6) = -----------------, F(6, 4, 7) = 0, F(6, 5, 0) = 0, 1440 42024249721 -142012317928177 F(6, 5, 1) = -----------, F(6, 5, 2) = 0, F(6, 5, 3) = ----------------, 10 180 4755998808282611 F(6, 5, 4) = 0, F(6, 5, 5) = ----------------, F(6, 5, 6) = 0, 120 -530248392067477699 50708745167 F(6, 5, 7) = -------------------, F(6, 6, 0) = -----------, F(6, 6, 1) = 0, 720 60 -43417261267913 4867997475680701 F(6, 6, 2) = ---------------, F(6, 6, 3) = 0, F(6, 6, 4) = ----------------, 120 144 -4864388749503364367 F(6, 6, 5) = 0, F(6, 6, 6) = --------------------, F(6, 6, 7) = 0, 4320 -2046854330783 F(6, 7, 0) = 0, F(6, 7, 1) = --------------, F(6, 7, 2) = 0, 30 305373605910129 F(6, 7, 3) = ---------------, F(6, 7, 4) = 0, 20 -68986784557575269 F(6, 7, 5) = ------------------, F(6, 7, 6) = 0, 72 361527755257880461151 F(6, 7, 7) = ---------------------, F(7, 0, 0) = 0, F(7, 0, 1) = 27/2, 15120 F(7, 0, 2) = 0, F(7, 0, 3) = 177201/4, F(7, 0, 4) = 0, 1164033207 2558149890417 F(7, 0, 5) = ----------, F(7, 0, 6) = 0, F(7, 0, 7) = -------------, 40 560 F(7, 1, 0) = 46/3, F(7, 1, 1) = 0, F(7, 1, 2) = 449272/3, F(7, 1, 3) = 0, 3037557153587 F(7, 1, 4) = 484060559/3, F(7, 1, 5) = 0, F(7, 1, 6) = -------------, 90 F(7, 1, 7) = 0, F(7, 2, 0) = 0, F(7, 2, 1) = 1020545/6, F(7, 2, 2) = 0, 4345556335 13119728132753 F(7, 2, 3) = ----------, F(7, 2, 4) = 0, F(7, 2, 5) = --------------, 12 120 -14527531092361451 F(7, 2, 6) = 0, F(7, 2, 7) = ------------------, F(7, 3, 0) = 64906, 5040 F(7, 3, 1) = 0, F(7, 3, 2) = 1233836876/3, F(7, 3, 3) = 0, -97346096577953 F(7, 3, 4) = 605010273791/3, F(7, 3, 5) = 0, F(7, 3, 6) = ---------------, 10 F(7, 3, 7) = 0, F(7, 4, 0) = 0, F(7, 4, 1) = 945488443/4, F(7, 4, 2) = 0, 16488396485473 -2636758507750885 F(7, 4, 3) = --------------, F(7, 4, 4) = 0, F(7, 4, 5) = -----------------, 72 144 4192492423980193729 F(7, 4, 6) = 0, F(7, 4, 7) = -------------------, F(7, 5, 0) = 274941793/5, 10080 F(7, 5, 1) = 0, F(7, 5, 2) = 800445994832/5, F(7, 5, 3) = 0, -1873796360616391 F(7, 5, 4) = -----------------, F(7, 5, 5) = 0, 90 28460340311569727 F(7, 5, 6) = -----------------, F(7, 5, 7) = 0, F(7, 6, 0) = 0, 36 1274556142717 -1717874674107199 F(7, 6, 1) = -------------, F(7, 6, 2) = 0, F(7, 6, 3) = -----------------, 20 120 129603855106618981 F(7, 6, 4) = 0, F(7, 6, 5) = ------------------, F(7, 6, 6) = 0, 144 -226649989806258319169 1167950789599 F(7, 6, 7) = ----------------------, F(7, 7, 0) = -------------, 10080 105 -576704699817254 F(7, 7, 1) = 0, F(7, 7, 2) = ----------------, F(7, 7, 3) = 0, 105 42996130956702429 F(7, 7, 4) = -----------------, F(7, 7, 5) = 0, 70 -96883903681826219783 F(7, 7, 6) = ---------------------, F(7, 7, 7) = 0 3780 -------------------------------------------- By the way, it took, 9.131, seconds to geneate the scheme. -------------------------------------------- Using this scheme we can compute any value very fast. For example F(1000, 2000, 3000) = -74091275435266458857731182729516610282934269066109395\ 823138458154131879442636344311523110510506202264490234209656703705793225\ 351955325253564408704668291062858039222935344187400448052411216422880109\ 880544726691898480545325521651910611240172992434291161385212489902689524\ 928751168699904154887089804520266781402212806534483430660089574213150977\ 001919771236634265274444020414990820285130746033130078890354114754125191\ 926147049957876031257708697304741140092148374268889773333198656955753629\ 202637403277508795971089086914861871642251741002734368006807492898096179\ 671839874417366776683676374062924639999057285489528047594841008889642983\ 446828681915380045031391554076732606325318435181388909412375068040303308\ 083686680503830009411840610637079731546831222567435982278811035417726418\ 795289695513375763325593881321035119642850231821469835288886160696688017\ 180742475969778186584455576258245261199767053881261253621981065277252434\ 648006817728648347467889356925810919427360364961165394258846976355852055\ 001314818910315639267129564905444708811132236213680587775812635011938041\ 386617441929913884845377078542523850780856938140391201815549422348779360\ 942621902010694290918507919233292310032419000139818769927228174249452788\ 681208655566879228837512842577390837164414813576593574348142186868798339\ 176228159511915574988782789975410663104491071942826897299533649420906565\ 764869476879083723885357296847487784604363879215774467321921786413630073\ 266326058243746652445839248094493756949886375938991168312241394734568606\ 600831995678425996559519737559599244897620841529925442925819018256153460\ 063049524007379194444416676969124548935915615297996047514405384725686689\ 336373072664425503557475254019635282202468441025457190900340802264291854\ 920265282333148790019709395270613710476954383828017859068620064517195674\ 833627169601483217408899465948263922741926171912034150125808169650292248\ 083722753665995612684897884219393959139810739301386845571121497138220079\ 089331724950390193572003584078342448981241009869364126212875177085868436\ 391127019896023537574574562758399519724176444265733312487169304607150712\ 837110392313197403970395147170295193743679947230501312967149513384626921\ 073194057681843022386100129968438114048004187942961969610850443763140060\ 196532700565157854929852744086082213818830125967097758743280798558364743\ 226265371004563856263237341796707577089111119013065777800219812701287943\ 906688356659805786343682531966811739969348987934590338967976121784727588\ 673397272095907709617274523257458021577670987341797631336419068996515122\ 569284922836243769413181059581943480920645528241725168300553445596642926\ 186135855005296442338633580275965504790206744196144310920735204572233783\ 146421856923469372480574549109974205291553792564724176947398450104999198\ 895200229238179269640254244543080044963279975009711750814611924074220832\ 499813675796110359917703431513468621209925586473765804163279299057683444\ 038363250171405989847246318555057257043114114703112180398048753494093817\ 873011616161092935544080328771179560009152165533607968520329719725623641\ 114772543740353043883452657169932316130405588334174219847292822262530600\ 856927938260226127106334025074060106502557404709888193666947245742752634\ 430941482262150302331590919603729515788432067861966575300272995168719757\ 901747632836110843097514586213916359554044903768602147241598094660803125\ 032750204563365210019741007921634036423891126611635446907252939865656366\ 434209246078598367934414684586134376601762735207191351993087653405048499\ 033442132300566701715628834363686751017878980767619823075501626627915828\ 452754611910802311121634372592479254337605754515458708293733562992361159\ 107650420030946148860550316703641903000296633375949538718811766814666470\ 724449232504018931774802714403376330315942725484869033282031808789414967\ 630496985626918648804292231191767458360137374085263648866697964820400340\ 054928381186551148667628957549785444158467412420814915518069465603868667\ 852947919006528242494949587005560897615915140937885440718692578140113488\ 306758296761836613975684254563483705321236446144982856493021214920530695\ 493085247237934753082335153196116802649355977155039339163521572290566096\ 944903070359754476038385251962288125256304720197067720394791593372244383\ 102299596724733448935453766334565896915847636102248337708218459840309168\ 646531447642853169547397570358831022307223120056141771046296520721169370\ 602491922113152009854755875341851843985817203354868946563676688374353245\ 112852479752306182634278269743997750505569370304705685373375842295718641\ 273721711527696287867993622296549102938141688178653561462500704062686916\ 887520310223768159947773169318635638379738324788692107889196124319310434\ 176467474744747906224734185160063577029386818203610356092375911147491811\ 574272791963985484383712628976400960888900128889522321885694845324648073\ 964365402073113288458210196874055469137869001017664029838219604566123808\ 882835023443200271664232226070925592456525430389297204216257722847194900\ 054184058371872165330092876368394050107725943549130405863997330462888784\ 674009519657024918687846628492849475995457526502614376849906864328674178\ 860349734548474682958657227908941202502591649142850466186245349241092286\ 741164573423776916882461667485357809985692969418823161797698469469299085\ 732282099546225342099302837504463885713906046216484994796198654158785476\ 334077828684396155120701480980550140933465615643388319317210005991957459\ 346613741080048649956131350801816767201435145206314443467252041972695448\ 997967273141643892030103198279394066823663201642777319889931733359117065\ 427719170260652729298994631178165487978109589435451812679949316907632873\ 297818188962337052135447341137012064453757015895273466473556544289944170\ 537984015368753733867893505669234086277318641902180769089587473375005468\ 792715892668214500087832192372197682530900209220054707710204436096528616\ / 0146294771248835790340974737196847 / 119756446289006117791439737878418\ / 909528166366656695098871328710727732123687558142863284668564083990155928\ 968041015158187157983456784847452135256363630554740417509547091062352718\ 726022467053032834496155190989700868240882686068296404387815731568110801\ 083961507164946488791096324725031715633997604168149382872581576463305811\ 707655341452559933889724699688792152013825499397652866619829044540817349\ 681551552492738956596536870233225969459745699750341451930604613238004340\ 522183381433370408561916896010172307443073291014839071979076748192799789\ 774361111546425779245382728741460043841036350598209884283564451718534725\ 024042609781660763012503854362814982172501845191834630148116412228492451\ 633418392192196457345281238333202824209273331728457660451400519584384678\ 561609480821024417415759539502159809014739212140784338910329335954226703\ 545192116221100144140965482981045732573536524336850341626599985090426891\ 416332470047271803513571478814522124743720359679000788321994108663068819\ 364675748264795498199348427534009605211577748161841097040320072492122977\ 110487242283611980014684915781235409943704602476296799344876476997481634\ 986777798337783560699139817143401897796645515841953913719087717094898050\ 933651483100432603596572728039342397383903445821359254127649163247381304\ 804873193200737822294180146962377814829019622932698474103853810367394992\ 227370773067010377583521668513918868524256210258691937347746780006023126\ 678230595763520368420735042620474106843516857065892391438315323448215321\ 096811977410106167315264846122117951933360145063679011087812477209340992\ 680270220657712511279639821550557563696385979751865585732737229197462541\ 156633070894806389193114035205325052477132172834860091980396442919575557\ 503080111601410455347706795358438651740464054888570774673408524430110667\ 029798289104388512678733099591639445603676596778398187081747127876513241\ 014586097552740231249285079192604858309818035938553047165708399952147581\ 867347011800990336177720412088365790617782549973736238520294228091360439\ 818441106922187338574103871142200352242006782083910696958378541979504616\ 603100062934925627054837147105436331923396696398578078876750652984507229\ 055059171801476352126798788305411789499173921708580556833578570383486143\ 561392067268041234114453988822166803449889877189066238793131417586602465\ 461715248412795564819864077611658444952082562192584939591148388688700745\ 453409008196604278719881982491694748469294351351206901212545748219462751\ 418870957094002592366492795724198983769308692001705273373637986102276759\ 216473137766287774864964027671036702285576515782298136062275659420797584\ 845295076598976133293311101358552978423715132467295860649435056250723990\ 253041154203069313126962027583960080137278948597139183702718441417335019\ 499613585058297503730509716293945941984741376377105623084974484986135210\ 542694089775894129374285836144950748385807053709556192217350976896498322\ 147363713605574462669015217668083184187199863252858876783142839483748794\ 831919363191872936393454118420304231557731117224452372409052202429633649\ 218534954065111561248247000840029201012036107844571810637024984916528451\ 881959814726848937824944097185669999125390337401894318181474485752941202\ 033482923376117731185086456991747823757991615410438017086378564571664385\ 235925141891743203355660997904980908408989098990746762286570181519603842\ 540435660037459392932132003028194699991732198056702868388674282288816390\ 627147356081044699222318647384683206755566419862306323938601784521346527\ 116811427711371201215692793859176292459437589322500970618944104248362993\ 221234598688297394972245201473114670027562564001837010590390769381259406\ 820719857511588816992118631301471096586477271633954457765395730848266045\ 953241442141821811840470555160067650063620364720155489013864866687392489\ 081752555822083540477085604367157696562799151605206598969515787791120876\ 318231914242283619119958442027933549098315260016625195075287731054304138\ 671214179167927307328487890457287899950369625076786283131830968307122593\ 919249120165115008660801299367624138513660767544231408303196053347877752\ 748529296724257349617775222503250975263326137654906235617056143195219089\ 530599066211529071171761987159575319054248740484708865767447695421514576\ 200276638218611407120048054158013530592830583323803172088778644943310476\ 648790638195044252960828666611038303159534219060697865577921434314113620\ 487023493499988364380082888867993973389136997150551947360255375275839394\ 204214862744650055866699837879166692167734957708219825434141931430806082\ 925799757988971199445865775429518439954256317462619218437267026471671303\ 928493629765757384855364510279648015992530104514573382401712157662592512\ 044345738158082011516989276695017024346200037590318770021025202562373703\ 954850282222456748905602734437260591359241317615230111864480622861855060\ 763031704637929624605834595138705225753091903292070503385466285134720194\ 368068580984816314307592312425004424574132331347625018885404459355969004\ 092174291402494026581538754260046288940291643812782910648164821862317804\ 340217303385572185720895582471118876712864115046498459559127016700769182\ 781884297360812021109504435614698639526472612905651111734322138286737996\ 268942240650812818673769526301782668723340221404389959279108086239709140\ 403107697076500666022558287990499423124062982745885280822130037585352045\ 831704504828092409739493310825046384696318117680434272030096398015427481\ 377432025885043139193159225761441489862728111898727450091655240098579136\ 684407603727350056255708133785171535646528194109704622536035713846393926\ 507688961807223970019771704603995580260663229918250696302178211188393972\ 952897674549695589143470607387520065847480805101448613235476626429744824\ 665136739588127626544071702819373178704322421819619903069603076586657591\ 027626160418249476653640682890942710597302020719543521447829698206240365\ 080937114850238796111618328409496125993846567370222832449644543561706585\ 599517933899177608020959868584557575240190288553078744617055969682216562\ 927000673134696878807625381905390280358816921946910127869096520988609911\ 594526570950370498040804083540718610921036601370799500523523016283076063\ 165296780797873306248493516688945209275386856677238345653635851514980688\ 487267084234949410234310951971442081467062339790048261109687118841331429\ 505893945314002025564270237463460529494003176315991086007945797546957633\ 570481038401778518330962893855438142509186889615676342331805942652432065\ 905045157834318352029917827082930732127712993360355285338454514410753235\ 624168368034297901186303489503866547727995110810549996592847130473025591\ 076843871152540779716350468139323453189559686917259175081104509424699529\ 799460099276873220339560727998613487487398434028399751155124067381549565\ 168063513405647140374749644972023936261392102140605254507249383025326246\ 567552602249612752724452375532847794783086981358163322466653068670617791\ 406117980523912636158948736929881642371992330778852080289865558969949749\ 673308596569109820594793722936374492903107019196521379080995864763891090\ 324976049479388538542831021309366868602457614016891477716494994565555475\ 700599235661248176711650717475618070480631011807609687383493031666715248\ 287083238023985543386708760447026611227333315045678567463615050105848457\ 823876074934148268277653850039025424510243180959207755436831064033536679\ 097452566070281500434664565752499162566816370203025977736180391140315352\ 491492866997641574935693530301487060829777215168187288244233083594791876\ 410717068770295321848847287520727545722344375993260650050691515308541721\ 904210651067156115129810755658821283072302031007602674996550421792327189\ 011407667702134839003095870182459724984365994372568778050998848743648065\ 104052838082890809479266304000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000000000000000000000\ 000000000000000000000000000000000000000000000000000000 This took, 39.450, seconds.