A187021 Coefficient of x^n in (1 + (n+1)*x + n*x^2)^n.

1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101

Offset

0,2

Links

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

Paul Barry and Aoife Hennessy, Generalized Narayana Polynomials, Riordan Arrays, and Lattice Paths, Journal of Integer Sequences, Vol. 15, 2012, #12.4.8. - N. J. A. Sloane, Oct 08 2012

Formula

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

a(n) = n^(n/2)*GegenbauerPoly(n,-n,-(n+1)/(2*sqrt(n)). - Emanuele Munarini, Oct 20 2016

a(n) = Sum_{k=0..n} binomial(n,k)^2 * n^k. - Paul D. Hanna, Mar 29 2011

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

a(n) = n! * [x^n] exp((n + 1)*x) * BesselI(0,2*sqrt(n)*x). - Ilya Gutkovskiy, May 31 2020

a(n) = hypergeom([-n, -n], [1], n). - Peter Luschny, Dec 22 2020

Maple

A187021:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -(n+1)/(2*sqrt(n))) );
1, seq(A187021(n), n = 1..30); # G. C. Greubel, May 31 2020
a := n -> hypergeom([-n, -n], [1], n):
seq(simplify(a(n)), n=0..18); # Peter Luschny, Dec 22 2020

Mathematica

Flatten[{1, Table[Sum[Binomial[n, k]^2*n^k, {k, 0, n}], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 17 2014 *)
Table[If[n==0, 1, Simplify[n^(n/2)*GegenbauerC[n, -n, -(n+1)/(2 Sqrt[n])]]], {n, 0, 30}] (* Emanuele Munarini, Oct 20 2016 *)

Prog

(PARI) {a(n)=sum(k=0, n, binomial(n, k)^2*n^k)} \\ Paul D. Hanna, Mar 29 2011
(Maxima) a(n):=coeff(expand((1+(n+1)*x+n*x^2)^n), x, n);
makelist(a(n), n, 0, 20);
(Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+(n+1)*x+n*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011
(Sage) [1]+[ n^(n/2)*gegenbauer(n, -n, -(n+1)/(2*sqrt(n))) for n in (1..30)] # G. C. Greubel, May 31 2020

Crossrefs

Main diagonal of A307883.

Cf. A092366, A186925, A187018, A187019, A241247, A234971.

Sequence in context: A047856 A246875 A242004 * A152059 A132063 A349291

Adjacent sequences: A187018 A187019 A187020 * A187022 A187023 A187024

Keyword

nonn

Author

Emanuele Munarini, Mar 02 2011

Status

approved