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

0,3

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)

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

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 *)

Prog

(PARI) q(n)=(1+n*x+n*x^2)^n; for(i=0, 20, print1(", "polcoeff(q(i), i)))
(Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+n*x+n*x^2)^n)[n+1]: n in [1..22] ]; // Klaus Brockhaus, Mar 03 2011
(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 */

Crossrefs

Cf. A186925, A187018.

Sequence in context: A318047 A338685 A373004 * A022519 A138439 A193563

Adjacent sequences: A092363 A092364 A092365 * A092367 A092368 A092369

Keyword

nonn

Author

Jon Perry, Mar 19 2004

Extensions

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

Status

approved