A092366 Coefficient of x^n in expansion of (1+n*x+n*x^2)^n.

1, 1, 8, 81, 1120, 19375, 400896, 9630411, 262955008, 8032730715, 271175200000, 10017828457483, 401738097475584, 17371952344599385, 805429080795852800, 39844314853048828125, 2094272851244149112832, 116526044312704751752451

Offset: 0

Author

Jon Perry, Mar 19 2004

Keywords

• nonn

Comments

Also coefficient of x^n in expansion of (1-2*n*x+(n^2-4*n)*x^2)^(-1/2). - Vladeta Jovovic, Mar 22 2004

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..381 (terms 1..100 from Vincenzo Librandi)

Crossrefs

Cf. A186925, A187018.

Programs

Magma

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

Maple

seq(n!*coeff(series(exp(n*x)*BesselI(0,2*sqrt(n)*x),x,n+1),x,n),n=1..17);

Mathematica

Table[Sum[n^k*Binomial[n,k]*Binomial[k,n-k],{k,Floor[n/2],n}],{n,1,20}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n == 0, 1, n^(n/2) GegenbauerC[n, -n, -Sqrt[n]/2]], {n, 0,
12}] (* Emanuele Munarini, Oct 20 2016 *)

Maxima

a(n):=coeff(expand((1+n*x+n*x^2)^n), x, n);
makelist(a(n), n, 1, 12); /* Emanuele Munarini, Mar 02 2011 */

PARI

q(n)=(1+n*x+n*x^2)^n; for(i=0,20,print1(","polcoeff(q(i),i)))

Formula

a(n) = n^(n/2)*GegenbauerPoly(n,-n,-sqrt(n)/2). - Emanuele Munarini, Oct 20 2016

Sum_{k=floor(n/2)..n} n^k*binomial(n, k)*binomial(k, n-k). - Vladeta Jovovic, Mar 22 2004

a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-2) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 17 2014

Extensions

a(0)=1 prepended by Seiichi Manyama, May 01 2019