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 A187019 Coefficient of x^n in expansion of (1+n*x+(n+1)*x^2)^n. 5
 1, 1, 10, 99, 1366, 23525, 484436, 11582375, 314830342, 9576682569, 322014499852, 11851803991115, 473634489404220, 20414267521982893, 943592267071798696, 46545155813085562575, 2439857423310573714758 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..100 FORMULA a(n) = [x^n] (1+n*x+(n+1)*x^2)^n. a(n) = Sum (C(n, k)*C(n-k, n-2*k)*n^(n-2*k)*(n+1)^k, k=0..floor(n/2)). a(n) ~ exp(2*sqrt(n)-2) * n^(n-1/4) / (2*sqrt(Pi)). - Vaclav Kotesovec, Apr 18 2014 a(n) = n! * [x^n] exp(n*x) * BesselI(0,2*sqrt(n + 1)*x). - Ilya Gutkovskiy, Jun 01 2020 MATHEMATICA Flatten[{1, Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^(n-2*k)*(n+1)^k, {k, 0, Floor[n/2]}], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 18 2014 *) Flatten[{1, Table[n^n * Hypergeometric2F1[1/2-n/2, -n/2, 1, 4*(1+n)/n^2], {n, 1, 20}]}] (* Vaclav Kotesovec, Apr 18 2014 *) PROG (Maxima) a(n):=coeff(expand((1+n*x+(n+1)*x^2)^n), x, n); makelist(a(n), n, 0, 12); (Magma) P:=PolynomialRing(Integers()); [ Coefficients((1+n*x+(n+1)*x^2)^n)[n+1]: n in [0..22] ]; // Klaus Brockhaus, Mar 03 2011 (PARI) a(n) = polcoef((1+n*x+(n+1)*x^2)^n, n); \\ Michel Marcus, Jun 01 2020 CROSSREFS Cf. A092366, A186925, A187018. Sequence in context: A000456 A138365 A190823 * A292429 A145644 A317055 Adjacent sequences: A187016 A187017 A187018 * A187020 A187021 A187022 KEYWORD nonn,easy AUTHOR Emanuele Munarini, Mar 02 2011 STATUS approved

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Last modified September 9 01:17 EDT 2024. Contains 375759 sequences. (Running on oeis4.)