On the coefficients of Psi(x,0) of the unique solution of the functional equ\ ation, 2 2 2 y Psi(x, y) + (x Psi(x, 0) y - y + x - y) Psi(x, y) + y - x Psi(x, 0) = 0 By Shalosh B. Ekhad Let's g(x)=Psi(x,0). The first, 24, (for the sake of the OEIS) are [1, 3, 13, 68, 399, 2530, 16965, 118668, 857956, 6369883, 48336171, 373537388, 2931682810, 23317105140, 187606350645, 1524813969276, 12504654858828, 103367824774012, 860593023907540, 7211115497448720, 60776550501588855, 514956972502029690, 4384387181372914755, 37495248874510995828] g=g(x) satisfies the following algebraic equation 4 3 3 2 2 2 2 g x + 3 g x + 8 g x + 3 g x - 20 g x + g + 16 x - 1 = 0 and in Maple notation g^4*x^3+3*g^3*x^2+8*g^2*x^2+3*g^2*x-20*g*x+g+16*x-1 = 0 More generally, psi=Psi(x,y) satisisfies the algebraic equation 8 8 7 8 7 6 7 7 6 8 6 6 psi y - 4 psi y + 4 psi x y - 7 psi y + 6 psi y - 4 psi x y 6 7 5 8 6 2 4 6 5 6 6 + 25 psi y - 4 psi y + 6 psi x y - 21 psi x y + 21 psi y 5 6 5 7 4 8 5 2 4 5 5 - 4 psi x y - 33 psi y + psi y + 4 psi x y + 42 psi x y 5 6 4 6 4 7 5 3 2 5 2 3 - 66 psi y + 4 psi x y + 19 psi y + 4 psi x y - 21 psi x y 5 4 5 5 4 2 4 4 5 4 6 + 45 psi x y - 35 psi y + 6 psi x y - psi x y + 75 psi y 3 7 4 3 2 4 2 3 4 4 4 5 - 4 psi y + 4 psi x y + 5 psi x y - 129 psi x y + 95 psi y 3 5 3 6 4 4 4 3 4 2 2 - 20 psi x y - 36 psi y + psi x - 7 psi x y + 27 psi x y 4 3 4 4 3 2 3 3 4 3 5 - 50 psi x y + 35 psi y - 28 psi x y + 47 psi x y - 90 psi y 2 6 3 3 3 2 2 3 3 3 4 + 6 psi y - 12 psi x y - 42 psi x y + 176 psi x y - 80 psi y 2 4 2 5 3 3 3 2 3 2 + 36 psi x y + 34 psi y + 3 psi x - 15 psi x y + 30 psi x y 3 3 2 2 2 2 3 2 4 5 - 21 psi y + 54 psi x y - 95 psi x y + 60 psi y - 4 psi y 2 3 2 2 2 2 2 3 3 + 8 psi x + 53 psi x y - 114 psi x y + 39 psi y - 28 psi x y 4 2 2 2 2 2 2 - 16 psi y + 3 psi x - 9 psi x y + 7 psi y - 48 psi x y 2 3 4 2 2 + 73 psi x y - 21 psi y + y - 20 psi x + 30 psi x y - 10 psi y 2 3 2 2 + 8 x y + 3 y + psi x - psi y + 16 x - 20 x y + 3 y - x + y = 0 and in Maple notation psi^8*y^8-4*psi^7*y^8+4*psi^7*x*y^6-7*psi^7*y^7+6*psi^6*y^8-4*psi^6*x*y^6+25* psi^6*y^7-4*psi^5*y^8+6*psi^6*x^2*y^4-21*psi^6*x*y^5+21*psi^6*y^6-4*psi^5*x*y^6 -33*psi^5*y^7+psi^4*y^8+4*psi^5*x^2*y^4+42*psi^5*x*y^5-66*psi^5*y^6+4*psi^4*x*y ^6+19*psi^4*y^7+4*psi^5*x^3*y^2-21*psi^5*x^2*y^3+45*psi^5*x*y^4-35*psi^5*y^5+6* psi^4*x^2*y^4-psi^4*x*y^5+75*psi^4*y^6-4*psi^3*y^7+4*psi^4*x^3*y^2+5*psi^4*x^2* y^3-129*psi^4*x*y^4+95*psi^4*y^5-20*psi^3*x*y^5-36*psi^3*y^6+psi^4*x^4-7*psi^4* x^3*y+27*psi^4*x^2*y^2-50*psi^4*x*y^3+35*psi^4*y^4-28*psi^3*x^2*y^3+47*psi^3*x* y^4-90*psi^3*y^5+6*psi^2*y^6-12*psi^3*x^3*y-42*psi^3*x^2*y^2+176*psi^3*x*y^3-80 *psi^3*y^4+36*psi^2*x*y^4+34*psi^2*y^5+3*psi^3*x^3-15*psi^3*x^2*y+30*psi^3*x*y^ 2-21*psi^3*y^3+54*psi^2*x^2*y^2-95*psi^2*x*y^3+60*psi^2*y^4-4*psi*y^5+8*psi^2*x ^3+53*psi^2*x^2*y-114*psi^2*x*y^2+39*psi^2*y^3-28*psi*x*y^3-16*psi*y^4+3*psi^2* x^2-9*psi^2*x*y+7*psi^2*y^2-48*psi*x^2*y+73*psi*x*y^2-21*psi*y^3+y^4-20*psi*x^2 +30*psi*x*y-10*psi*y^2+8*x*y^2+3*y^3+psi*x-psi*y+16*x^2-20*x*y+3*y^2-x+y = 0 writing Psi(x,0)=g(x) as a Taylor series around x=0 infinity ----- \ n g(x) = ) a[n] x / ----- n = 0 The coefficients, a[n], satisfy the folllowing linear recurrence equation wi\ th polynomial coefficients of order, 1 (4 n + 5) (2 n + 1) (4 n + 3) a(n) -8/3 ---------------------------------- + a(n + 1) = 0 (3 n + 5) (3 n + 4) (n + 2) and in Maple notation -8/3*(4*n+5)*(2*n+1)*(4*n+3)/(3*n+5)/(3*n+4)/(n+2)*a(n)+a(n+1) = 0 subject to the initial conditions a(1) = 1 Finally, just for fun here is , a(1000) a(1000) = 971916972145580196581205042601861574138413467747362447080338099405602\ 9162669274726393496719280270022594298094562195877640239929144027886626786691614\ 2234143409385374189366908017496445184612454002512315167249839366054341707603615\ 3742058138630673617108025947188088221111036120585787618962749626045188277322621\ 1686943757594036220331346247068140687625106858467405310844840243673170498555984\ 0846056190098041661136830643103170376050029595863088814657375283724627793962753\ 3048879524040960753518275546646776065186350474294157663112857446575256876476039\ 9341953716271369358839165583357970573724602308665314705698865652997358027045788\ 6455329101727036959875035313710112846291113394632798119389519842314851621629257\ 2390127114692899868226861222160103802304286679126223053389370834411774779879190\ 7105511110893675456111697013440561859023120949404994925771013253355415344271278\ 5507635399790714017081687623439728329424486618725545549301607739045412850548053\ 9294825754857889882250067580800 --------------------------------------- This ends this paper that took, 0.772, seconds to generate