A324956 Triangle of coefficients T(n,k) of y^n in Product_{k=0..n-2} (n + (n + k)*y + n*y^2), as read by rows of terms k = 0..2*n-2, for n >= 1.
1, 2, 2, 2, 9, 21, 30, 21, 9, 64, 240, 488, 600, 488, 240, 64, 625, 3250, 8775, 15080, 17980, 15080, 8775, 3250, 625, 7776, 51840, 176040, 387360, 606384, 701280, 606384, 387360, 176040, 51840, 7776, 117649, 957999, 3935239, 10557540, 20437361, 29924601, 33904822, 29924601, 20437361, 10557540, 3935239, 957999, 117649, 2097152, 20185088, 97484800, 308768768, 711782400, 1258039552, 1753753728, 1956209024, 1753753728, 1258039552, 711782400, 308768768, 97484800, 20185088, 2097152
Offset: 1
Examples
E.g.f.: A(x,y) = Sum_{n>=0} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k starts
A(x) = x + (2*y^2 + 2*y + 2)*x^2/2! + (9*y^4 + 21*y^3 + 30*y^2 + 21*y + 9)*x^3/3! + (64*y^6 + 240*y^5 + 488*y^4 + 600*y^3 + 488*y^2 + 240*y + 64)*x^4/4! + (625*y^8 + 3250*y^7 + 8775*y^6 + 15080*y^5 + 17980*y^4 + 15080*y^3 + 8775*y^2 + 3250*y + 625)*x^5/5! + (7776*y^10 + 51840*y^9 + 176040*y^8 + 387360*y^7 + 606384*y^6 + 701280*y^5 + 606384*y^4 + 387360*y^3 + 176040*y^2 + 51840*y + 7776)*x^6/6! + (117649*y^12 + 957999*y^11 + 3935239*y^10 + 10557540*y^9 + 20437361*y^8 + 29924601*y^7 + 33904822*y^6 + 29924601*y^5 + 20437361*y^4 + 10557540*y^3 + 3935239*y^2 + 957999*y + 117649)*x^7/7! + (2097152*y^14 + 20185088*y^13 + 97484800*y^12 + 308768768*y^11 + 711782400*y^10 + 1258039552*y^9 + 1753753728*y^8 + 1956209024*y^7 + 1753753728*y^6 + 1258039552*y^5 + 711782400*y^4 + 308768768*y^3 + 97484800*y^2 + 20185088*y + 2097152)*x^8/8! + ...
This triangle of coefficients T(n,k) of x^n*y^k/n! in A(x,y) begins:
1;
2, 2, 2;
9, 21, 30, 21, 9;
64, 240, 488, 600, 488, 240, 64;
625, 3250, 8775, 15080, 17980, 15080, 8775, 3250, 625;
7776, 51840, 176040, 387360, 606384, 701280, 606384, 387360, 176040, 51840, 7776;
117649, 957999, 3935239, 10557540, 20437361, 29924601, 33904822, 29924601, 20437361, 10557540, 3935239, 957999, 117649;
2097152, 20185088, 97484800, 308768768, 711782400, 1258039552, 1753753728, 1956209024, 1753753728, 1258039552, 711782400, 308768768, 97484800, 20185088, 2097152; ...
Programs
-
PARI
{T(n, k) = polcoeff(prod(m=0, n-2, n + (n+m)*y + n*y^2 +y*O(y^k)), k, y)} for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print("")) -
PARI
{T(n,k) = my(A = serreverse( x*(1 - x*y +x*O(x^n) )^(1/y+1+y))); n!*polcoeff(polcoeff(A,n,x),k,y)} for(n=1, 10, for(k=0, 2*n-2, print1(T(n, k), ", ")); print(""))
Formula
E.g.f. A(x) = Sum_{n>=1} x^n/n! * Sum_{k=0..2*n-2} T(n,k)*y^k satisfies
(1) A(x,y) = Sum_{n>=1} x^n/n! * Product_{k=0..n-2} (n + (n + k)*y + n*y^2).
(2) A(x,y) = Series_Reversion( x*(1 - x*y)^(1/y+1+y) ).
(3) A(x,y) = x/(1 - y*A(x))^(1/y+1+y).
(4) A(x,y) = x*Sum_{n>=0} A(x,y)^n/n! * Product_{k=0..n-1} (1 + (k+1)*y + y^2).
PARTICULAR ARGUMENTS.
E.g.f. at y = 0: A(x,y=0) = -LambertW(-x) = x*exp(-LambertW(-x)).
E.g.f. at y = 1: A(x,y=1) = x*G(x)^3, where G(x) = 1 + x*G(x)^4 is the g.f. of A002293.
FORMULAS INVOLVING TERMS.
Row sums: Sum_{k=0..2*n-2} T(n,k) = (4*n-2)!/(3*n-1)! for n >= 1.
T(n,0) = T(n,2*n-2) = n^(n-1) for n >= 1.
T(n,n-1) = A324957(n) for n >= 1.