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A187021 Coefficient of x^n in (1+(n+1)*x+n*x^2)^n. 9
1, 2, 13, 136, 1921, 33876, 712909, 17383584, 481003009, 14869654300, 507406003501, 18928740714192, 765897591633409, 33392080668673832, 1559976990077534253, 77717020110946293376, 4111810085670587224065, 230190619432401207833004, 13591965974806603671569101 (list; graph; refs; listen; history; text; internal format)
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

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, 100}] (* 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, 12);

(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

CROSSREFS

Main diagonal of A307883.

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

Sequence in context: A047856 A246875 A242004 * A152059 A132063 A318003

Adjacent sequences:  A187018 A187019 A187020 * A187022 A187023 A187024

KEYWORD

nonn

AUTHOR

Emanuele Munarini, Mar 02 2011

STATUS

approved

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Last modified October 22 22:34 EDT 2019. Contains 328335 sequences. (Running on oeis4.)