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 A324958 #16 Oct 30 2019 13:27:17
%S A324958 1,2,4,2,9,39,60,39,9,64,432,1160,1584,1160,432,64,625,5750,22275,
%T A324958 47380,60460,47380,22275,5750,625,7776,90720,461160,1343160,2479464,
%U A324958 3029040,2479464,1343160,461160,90720,7776,117649,1663893,10489969,38937360,94679711,158760987,188149822,158760987,94679711,38937360,10489969,1663893,117649,2097152,34865152,262635520,1187049472,3593318400,7701010688,12043471488,13957194496,12043471488,7701010688,3593318400,1187049472,262635520,34865152,2097152
%N A324958 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.
%F A324958 E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
%F A324958 (1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (2*n + k)*y + n*y^2).
%F A324958 (2) A(x,y) = Series_Reversion( x*(1 - x*y)^((1+y)^2/y) ), wrt x.
%F A324958 (3) A(x,y) = x/(1 - y*A(x))^((1+y)^2/y).
%F A324958 (4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+2)*y + y^2).
%F A324958 PARTICULAR ARGUMENTS.
%F A324958 E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
%F A324958 E.g.f. at y = 1: A(x,y=1) = x*G(x)^4, where G(x) = 1 + x*G(x)^5 is the g.f. of A002294.
%F A324958 FORMULAS INVOLVING TERMS.
%F A324958 Row sums: Sum_{k=0..2*n-2} T(n,k) = (5*n-2)!/(4*n-1)! for n >= 1.
%F A324958 T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
%F A324958 T(n,n-1) = A324959(n) for n >= 1.
%e A324958 E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
%e A324958 A(x,y) = x + (2*y^2 + 4*y + 2)*x^2/2! + (9*y^4 + 39*y^3 + 60*y^2 + 39*y + 9)*x^3/3! + (64*y^6 + 432*y^5 + 1160*y^4 + 1584*y^3 + 1160*y^2 + 432*y + 64)*x^4/4! + (625*y^8 + 5750*y^7 + 22275*y^6 + 47380*y^5 + 60460*y^4 + 47380*y^3 + 22275*y^2 + 5750*y + 625)*x^5/5! + (7776*y^10 + 90720*y^9 + 461160*y^8 + 1343160*y^7 + 2479464*y^6 + 3029040*y^5 + 2479464*y^4 + 1343160*y^3 + 461160*y^2 + 90720*y + 7776)*x^6/6! + (117649*y^12 + 1663893*y^11 + 10489969*y^10 + 38937360*y^9 + 94679711*y^8 + 158760987*y^7 + 188149822*y^6 + 158760987*y^5 + 94679711*y^4 + 38937360*y^3 + 10489969*y^2 + 1663893*y + 117649)*x^7/7! + (2097152*y^14 + 34865152*y^13 + 262635520*y^12 + 1187049472*y^11 + 3593318400*y^10 + 7701010688*y^9 + 12043471488*y^8 + 13957194496*y^7 + 12043471488*y^6 + 7701010688*y^5 + 3593318400*y^4 + 1187049472*y^3 + 262635520*y^2 + 34865152*y + 2097152)*x^8/8! + ...
%e A324958 This triangle of coefficients T(n,k) of x^n*y^k/n! in e.g.f. A(x,y) begins:
%e A324958 1;
%e A324958 2, 4, 2;
%e A324958 9, 39, 60, 39, 9;
%e A324958 64, 432, 1160, 1584, 1160, 432, 64;
%e A324958 625, 5750, 22275, 47380, 60460, 47380, 22275, 5750, 625;
%e A324958 7776, 90720, 461160, 1343160, 2479464, 3029040, 2479464, 1343160, 461160, 90720, 7776;
%e A324958 117649, 1663893, 10489969, 38937360, 94679711, 158760987, 188149822, 158760987, 94679711, 38937360, 10489969, 1663893, 117649;
%e A324958 2097152, 34865152, 262635520, 1187049472, 3593318400, 7701010688, 12043471488, 13957194496, 12043471488, 7701010688, 3593318400, 1187049472, 262635520, 34865152, 2097152; ...
%o A324958 (PARI) {T(n, k) = polcoeff(prod(m=0, n-2, n + (2*n+m)*y + n*y^2 +y*O(y^k)), k, y)}
%o A324958 for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
%o A324958 (PARI) {T(n,k) = my(A = serreverse( x*(1 - x*y +x*O(x^n) )^((1+y)^2/y)));
%o A324958 n!*polcoeff(polcoeff(A,n,x),k,y)}
%o A324958 for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
%Y A324958 Cf. A324959, A002294.
%Y A324958 Cf. A324960, A324305.
%K A324958 nonn,tabf
%O A324958 1,2
%A A324958 _Paul D. Hanna_, Mar 20 2019