A187019 Coefficient of x^n in expansion of (1+n*x+(n+1)*x^2)^n.

1, 1, 10, 99, 1366, 23525, 484436, 11582375, 314830342, 9576682569, 322014499852, 11851803991115, 473634489404220, 20414267521982893, 943592267071798696, 46545155813085562575, 2439857423310573714758

Offset: 0

Author

Emanuele Munarini, Mar 02 2011

Keywords

• nonn
• easy

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..100

Crossrefs

Cf. A092366, A186925, A187018.

Programs

Magma

P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+(n+1)*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011

Mathematica

Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k)*(n+1)^k, {k,0,Floor[n/2]}],{n,1,20}]}] (* Vaclav Kotesovec, Apr 18 2014 *)
Flatten[{1,Table[n^n * Hypergeometric2F1[1/2-n/2,-n/2,1,4*(1+n)/n^2],{n,1,20}]}] (* Vaclav Kotesovec, Apr 18 2014 *)

Maxima

a(n):=coeff(expand((1+n*x+(n+1)*x^2)^n),x,n);
makelist(a(n),n,0,12);

PARI

a(n) = polcoef((1+n*x+(n+1)*x^2)^n, n); \\ Michel Marcus, Jun 01 2020

Formula

a(n) = [x^n] (1+n*x+(n+1)*x^2)^n.

a(n) = Sum (C(n, k)*C(n-k, n-2*k)*n^(n-2*k)*(n+1)^k, k=0..floor(n/2)).

a(n) ~ exp(2*sqrt(n)-2) * n^(n-1/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 18 2014

a(n) = n! * [x^n] exp(n*x) * BesselI(0,2*sqrt(n + 1)*x). - Ilya Gutkovskiy, Jun 01 2020