A186925 Coefficient of x^n in (1+n*x+x^2)^n.

1, 1, 6, 45, 454, 5775, 88796, 1602447, 33213510, 777665691, 20302315252, 584774029983, 18422140045596, 630132567760345, 23257790717110392, 921362075184792825, 38994274473840538182, 1755943506127367745795, 83829045032101462204100, 4229207755493569286374167

Offset

0,3

Links

Seiichi Manyama, Table of n, a(n) for n = 0..386 (terms 0..100 from Vincenzo Librandi)

Formula

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

a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^(n-2*k).

a(n) ~ BesselI(0,2) * n^n. - Vaclav Kotesovec, Apr 17 2014

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

From Ilya Gutkovskiy, Sep 20 2017: (Start)

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

a(n) = n! * [x^n] exp(n*x)*BesselI(0,2*x). (End)

From Seiichi Manyama, May 01 2019: (Start)

a(n) = Sum_{k=0..n} (n-2)^(n-k) * binomial(n,k) * binomial(2*k,k).

a(n) = Sum_{k=0..n} (n+2)^(n-k) * (-1)^k * binomial(n,k) * binomial(2*k,k). (End)

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k), {k, 0, Floor[n/2]}], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[GegenbauerC[n, -n, -n/2] + KroneckerDelta[n, 0], {n, 0, 100}] (* Emanuele Munarini, Oct 20 2016 *)
Table[SeriesCoefficient[(1 + n*x + x^2)^n, {x, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 13 2023 *)

Prog

(Maxima) a(n):=coeff(expand((1+n*x+x^2)^n), x, n);
(Maxima) makelist(ultraspherical(n, -n, -n/2), n, 0, 12); /* Emanuele Munarini, Oct 20 2016 */
makelist(a(n), n, 0, 20);
(Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+n*x+x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 02 2011
(PARI) {a(n) = sum(k=0, n, (n-2)^(n-k)*binomial(n, k)*binomial(2*k, k))} \\ Seiichi Manyama, May 01 2019
(PARI) a(n) = polcoef((1+n*x+x^2)^n, n); \\ Michel Marcus, May 01 2019

Crossrefs

Main diagonal of A292627.

Cf. A092366, A187018, A187019, A187021, A070910, A292629.

Sequence in context: A228194 A331726 A084064 * A368793 A294642 A346682

Adjacent sequences: A186922 A186923 A186924 * A186926 A186927 A186928

Keyword

nonn,easy

Author

Emanuele Munarini, Mar 02 2011

Status

approved