This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A324960 #16 May 28 2025 08:55:10
%S A324960 1,1,2,1,1,5,8,5,1,1,9,29,42,29,9,1,1,14,75,196,268,196,75,14,1,1,20,
%T A324960 160,660,1519,2000,1519,660,160,20,1,1,27,301,1800,6299,13293,17038,
%U A324960 13293,6299,1800,301,27,1,1,35,518,4235,21000,65485,129681,162890,129681,65485,21000,4235,518,35,1,1,44,834,8932,59633,258720,740046,1395504,1725372,1395504,740046,258720,59633,8932,834,44,1,1,54,1275,17316,149787,863982,3386879,9054684,16420458,20044728,16420458,9054684,3386879,863982,149787,17316,1275,54,1
%N A324960 Triangle of coefficients T(n,k) of y^k in Product_{k=0..n-1} (1 + (k+2)*y + y^2), read by rows of terms k = 0..2*n, for n >= 0.
%F A324960 E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k satisfies
%F A324960 (1) A(x,y) = Sum_{n>=0} x^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
%F A324960 (2) A(x,y) = 1/(1 - x*y)^((1+y)^2/y).
%F A324960 (3) x = Sum_{n>=1} (x/A(x,y))^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
%F A324960 Row sums are (n+3)!/3! for row n >= 0.
%e A324960 E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n} T(n,k)*y^k starts
%e A324960 A(x,y) = 1 + (y^2 + 2*y + 1)*x + (y^4 + 5*y^3 + 8*y^2 + 5*y + 1)*x^2/2! + (y^6 + 9*y^5 + 29*y^4 + 42*y^3 + 29*y^2 + 9*y + 1)*x^3/3! + (y^8 + 14*y^7 + 75*y^6 + 196*y^5 + 268*y^4 + 196*y^3 + 75*y^2 + 14*y + 1)*x^4/4! + (y^10 + 20*y^9 + 160*y^8 + 660*y^7 + 1519*y^6 + 2000*y^5 + 1519*y^4 + 660*y^3 + 160*y^2 + 20*y + 1)*x^5/5! + (y^12 + 27*y^11 + 301*y^10 + 1800*y^9 + 6299*y^8 + 13293*y^7 + 17038*y^6 + 13293*y^5 + 6299*y^4 + 1800*y^3 + 301*y^2 + 27*y + 1)*x^6/6! + (y^14 + 35*y^13 + 518*y^12 + 4235*y^11 + 21000*y^10 + 65485*y^9 + 129681*y^8 + 162890*y^7 + 129681*y^6 + 65485*y^5 + 21000*y^4 + 4235*y^3 + 518*y^2 + 35*y + 1)*x^7/7! + (y^16 + 44*y^15 + 834*y^14 + 8932*y^13 + 59633*y^12 + 258720*y^11 + 740046*y^10 + 1395504*y^9 + 1725372*y^8 + 1395504*y^7 + 740046*y^6 + 258720*y^5 + 59633*y^4 + 8932*y^3 + 834*y^2 + 44*y + 1)*x^8/8! + ...
%e A324960 This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
%e A324960 1;
%e A324960 1, 2, 1;
%e A324960 1, 5, 8, 5, 1;
%e A324960 1, 9, 29, 42, 29, 9, 1;
%e A324960 1, 14, 75, 196, 268, 196, 75, 14, 1;
%e A324960 1, 20, 160, 660, 1519, 2000, 1519, 660, 160, 20, 1;
%e A324960 1, 27, 301, 1800, 6299, 13293, 17038, 13293, 6299, 1800, 301, 27, 1;
%e A324960 1, 35, 518, 4235, 21000, 65485, 129681, 162890, 129681, 65485, 21000, 4235, 518, 35, 1;
%e A324960 1, 44, 834, 8932, 59633, 258720, 740046, 1395504, 1725372, 1395504, 740046, 258720, 59633, 8932, 834, 44, 1;
%e A324960 1, 54, 1275, 17316, 149787, 863982, 3386879, 9054684, 16420458, 20044728, 16420458, 9054684, 3386879, 863982, 149787, 17316, 1275, 54, 1; ...
%o A324960 (PARI) {T(n, k) = polcoeff( prod(m=0, n-1, 1 + (m+2)*y + y^2 +x*O(x^k)), k, y)}
%o A324960 for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))
%o A324960 (PARI) {T(n, k) = n!*polcoeff(polcoeff( 1/(1 - x*y +x*O(x^n) )^((1+y)^2/y),n, x), k, y)}
%o A324960 for(n=0, 10, for(k=0, 2*n, print1(T(n, k), ", ")); print(""))
%Y A324960 Cf. A324961, A324962.
%Y A324960 Cf. A324958, A249790.
%K A324960 nonn,tabf
%O A324960 0,3
%A A324960 _Paul D. Hanna_, Mar 20 2019