Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 3, [a[1], a[2], a[3]] satisfing the set of conditions {a[1] <= a[2] + a[3]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n 1 ) c(n) t = - --------------------------------------- / 2 2 2 3 ----- (t + 1) (t + t + 1) (t + 1) (t - 1) n = 0 and in Maple foramt it is -1/(t^2+1)/(t^2+t+1)/(t+1)^2/(t-1)^3 -------------------------------- This ends this theorem that took, 0.084, seconds. Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 3, [a[1], a[2], a[3]] satisfing the set of conditions {a[1] <= 2 a[2], a[2] <= 2 a[3]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n 9 8 7 6 5 4 3 2 / ) c(n) t = - (t + 2 t + 2 t + 2 t + t + t + t + t + t + 1) / / / ----- n = 0 2 4 3 2 ((t + 1) (t + t + 1) (t + t + t + t + 1) 6 5 4 3 2 2 3 (t + t + t + t + t + t + 1) (t + 1) (t - 1) ) and in Maple foramt it is -(t^9+2*t^8+2*t^7+2*t^6+t^5+t^4+t^3+t^2+t+1)/(t+1)/(t^2+t+1)/(t^4+t^3+t^2+t+1)/ (t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+1)/(t-1)^3 -------------------------------- This ends this theorem that took, 0.216, seconds. Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 3, [a[1], a[2], a[3]] satisfing the set of conditions {a[1] <= 3 a[2], a[2] <= 3 a[3]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n 20 19 18 17 16 15 14 ) c(n) t = - (t + 1) (t + t + 2 t + t + 2 t + t + 3 t / ----- n = 0 13 12 11 10 9 8 7 6 5 4 + 2 t + 4 t + 3 t + 5 t + 4 t + 5 t + 2 t + 3 t + t + 2 t 2 / 2 4 3 2 + t + 1) / ((t + t + 1) (t + t + t + t + 1) / 6 5 4 3 2 (t + t + t + t + t + t + 1) 12 11 10 9 8 7 6 5 4 3 2 3 (t + t + t + t + t + t + t + t + t + t + t + t + 1) (t - 1) ) and in Maple foramt it is -(t+1)*(t^20+t^19+2*t^18+t^17+2*t^16+t^15+3*t^14+2*t^13+4*t^12+3*t^11+5*t^10+4* t^9+5*t^8+2*t^7+3*t^6+t^5+2*t^4+t^2+1)/(t^2+t+1)/(t^4+t^3+t^2+t+1)/(t^6+t^5+t^4 +t^3+t^2+t+1)/(t^12+t^11+t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t-1)^3 -------------------------------- This ends this theorem that took, 0.926, seconds. Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 4, [a[1], a[2], a[3], a[4]] satisfing the set of conditions {a[1] <= a[2] + a[3], a[2] <= a[3] + a[4]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n ) c(n) t = ( / ----- n = 0 10 9 8 7 6 5 4 3 2 / t + 2 t + 2 t + 2 t + 2 t + 2 t + 2 t + t + t + t + 1) / ( / 2 4 3 2 6 5 4 3 2 (t - t + 1) (t + t + t + t + 1) (t + t + t + t + t + t + 1) 2 2 2 2 4 (t + 1) (t + 1) (t + t + 1) (t - 1) ) and in Maple foramt it is (t^10+2*t^9+2*t^8+2*t^7+2*t^6+2*t^5+2*t^4+t^3+t^2+t+1)/(t^2-t+1)/(t^4+t^3+t^2+t +1)/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+1)/(t+1)^2/(t^2+t+1)^2/(t-1)^4 -------------------------------- This ends this theorem that took, 1.448, seconds. Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 4, [a[1], a[2], a[3], a[4]] satisfing the set of conditions {a[1] <= 2 a[2], a[2] <= 2 a[3], a[3] <= 2 a[4]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n 38 37 36 35 34 33 32 ) c(n) t = (t + 2 t + 4 t + 6 t + 9 t + 13 t + 16 t / ----- n = 0 31 30 29 28 27 26 25 24 + 20 t + 24 t + 27 t + 31 t + 34 t + 37 t + 40 t + 44 t 23 22 21 20 19 18 17 16 + 47 t + 49 t + 52 t + 53 t + 53 t + 54 t + 51 t + 47 t 15 14 13 12 11 10 9 8 + 43 t + 38 t + 33 t + 27 t + 21 t + 17 t + 13 t + 10 t 7 6 5 4 3 2 / 2 + 7 t + 5 t + 4 t + 3 t + 2 t + t + t + 1) / ((t - t + 1) / 4 3 2 6 5 4 3 2 (t + t + t + t + 1) (t + t + t + t + t + t + 1) 10 9 8 7 6 5 4 3 2 2 (t + t + t + t + t + t + t + t + t + t + 1) (t + 1) 8 7 5 4 3 6 3 4 2 (t - t + t - t + t - t + 1) (t + t + 1) (t + 1) (t + 1) 2 2 4 (t + t + 1) (t - 1) ) and in Maple foramt it is (t^38+2*t^37+4*t^36+6*t^35+9*t^34+13*t^33+16*t^32+20*t^31+24*t^30+27*t^29+31*t^ 28+34*t^27+37*t^26+40*t^25+44*t^24+47*t^23+49*t^22+52*t^21+53*t^20+53*t^19+54*t ^18+51*t^17+47*t^16+43*t^15+38*t^14+33*t^13+27*t^12+21*t^11+17*t^10+13*t^9+10*t ^8+7*t^7+5*t^6+4*t^5+3*t^4+2*t^3+t^2+t+1)/(t^2-t+1)/(t^4+t^3+t^2+t+1)/(t^6+t^5+ t^4+t^3+t^2+t+1)/(t^10+t^9+t^8+t^7+t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+1)/(t^8-t^7+t^ 5-t^4+t^3-t+1)/(t^6+t^3+1)/(t^4+1)/(t+1)^2/(t^2+t+1)^2/(t-1)^4 -------------------------------- This ends this theorem that took, 19.075, seconds. Theorem : Let c(n) be the number of weakly-decreasing arrays of non-negative integers \ of length, 5, [a[1], a[2], a[3], a[4], a[5]] satisfing the set of conditions {a[1] <= a[2] + a[3], a[2] <= a[3] + a[4], a[3] <= a[4] + a[5]} Then the ordinary generating function of c(n) is given by the following rati\ onal function infinity ----- \ n 22 21 20 19 18 17 16 15 ) c(n) t = - (t + t + 2 t + t + 2 t + t + 2 t + t / ----- n = 0 14 13 12 11 10 9 8 7 6 5 + 3 t + 2 t + 3 t + 2 t + 3 t + 2 t + 2 t + t + 2 t + t 4 / 2 2 4 3 2 4 + t + 1) / ((t + t + 1) (t - t + 1) (t - t + t - t + 1) (t + 1) / 4 2 6 5 4 3 2 2 2 (t - t + 1) (t + t + t + t + t + t + 1) (t + 1) 4 3 2 2 3 5 (t + t + t + t + 1) (t + 1) (t - 1) ) and in Maple foramt it is -(t^22+t^21+2*t^20+t^19+2*t^18+t^17+2*t^16+t^15+3*t^14+2*t^13+3*t^12+2*t^11+3*t ^10+2*t^9+2*t^8+t^7+2*t^6+t^5+t^4+1)/(t^2+t+1)/(t^2-t+1)/(t^4-t^3+t^2-t+1)/(t^4 +1)/(t^4-t^2+1)/(t^6+t^5+t^4+t^3+t^2+t+1)/(t^2+1)^2/(t^4+t^3+t^2+t+1)^2/(t+1)^3 /(t-1)^5 -------------------------------- This ends this theorem that took, 24.127, seconds.