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%I #32 Sep 08 2022 08:45:13
%S 1,1,8,81,1120,19375,400896,9630411,262955008,8032730715,271175200000,
%T 10017828457483,401738097475584,17371952344599385,805429080795852800,
%U 39844314853048828125,2094272851244149112832,116526044312704751752451
%N Coefficient of x^n in expansion of (1+n*x+n*x^2)^n.
%C Also coefficient of x^n in expansion of (1-2*n*x+(n^2-4*n)*x^2)^(-1/2). - _Vladeta Jovovic_, Mar 22 2004
%H Seiichi Manyama, <a href="/A092366/b092366.txt">Table of n, a(n) for n = 0..381</a> (terms 1..100 from Vincenzo Librandi)
%F a(n) = n^(n/2)*GegenbauerPoly(n,-n,-sqrt(n)/2). - _Emanuele Munarini_, Oct 20 2016
%F Sum_{k=floor(n/2)..n} n^k*binomial(n, k)*binomial(k, n-k). - _Vladeta Jovovic_, Mar 22 2004
%F a(n) ~ n^(n-1/4) * exp(2*sqrt(n)-2) / (2*sqrt(Pi)). - _Vaclav Kotesovec_, Apr 17 2014
%p seq(n!*coeff(series(exp(n*x)*BesselI(0,2*sqrt(n)*x),x,n+1),x,n),n=1..17);
%t Table[Sum[n^k*Binomial[n,k]*Binomial[k,n-k],{k,Floor[n/2],n}],{n,1,20}] (* _Vaclav Kotesovec_, Apr 17 2014 *)
%t Table[If[n == 0, 1, n^(n/2) GegenbauerC[n, -n, -Sqrt[n]/2]], {n, 0,
%t 12}] (* _Emanuele Munarini_, Oct 20 2016 *)
%o (PARI) q(n)=(1+n*x+n*x^2)^n; for(i=0,20,print1(","polcoeff(q(i),i)))
%o (Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+n*x+n*x^2)^n)[n+1]: n in [1..22] ]; // _Klaus Brockhaus_, Mar 03 2011
%o (Maxima) a(n):=coeff(expand((1+n*x+n*x^2)^n), x, n);
%o makelist(a(n), n, 1, 12); /* _Emanuele Munarini_, Mar 02 2011 */
%Y Cf. A186925, A187018.
%K nonn
%O 0,3
%A _Jon Perry_, Mar 19 2004
%E a(0)=1 prepended by _Seiichi Manyama_, May 01 2019
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