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%I #26 Sep 08 2022 08:45:56
%S 1,1,10,99,1366,23525,484436,11582375,314830342,9576682569,
%T 322014499852,11851803991115,473634489404220,20414267521982893,
%U 943592267071798696,46545155813085562575,2439857423310573714758
%N Coefficient of x^n in expansion of (1+n*x+(n+1)*x^2)^n.
%H Vincenzo Librandi, <a href="/A187019/b187019.txt">Table of n, a(n) for n = 0..100</a>
%F a(n) = [x^n] (1+n*x+(n+1)*x^2)^n.
%F a(n) = Sum (C(n, k)*C(n-k, n-2*k)*n^(n-2*k)*(n+1)^k, k=0..floor(n/2)).
%F a(n) ~ exp(2*sqrt(n)-2) * n^(n-1/4) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Apr 18 2014
%F a(n) = n! * [x^n] exp(n*x) * BesselI(0,2*sqrt(n + 1)*x). - _Ilya Gutkovskiy_, Jun 01 2020
%t 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 *)
%t 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 *)
%o (Maxima) a(n):=coeff(expand((1+n*x+(n+1)*x^2)^n),x,n);
%o makelist(a(n),n,0,12);
%o (Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+n*x+(n+1)*x^2)^n)[n+1]: n in [0..22] ]; // _Klaus Brockhaus_, Mar 03 2011
%o (PARI) a(n) = polcoef((1+n*x+(n+1)*x^2)^n, n); \\ _Michel Marcus_, Jun 01 2020
%Y Cf. A092366, A186925, A187018.
%K nonn,easy
%O 0,3
%A _Emanuele Munarini_, Mar 02 2011
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