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A187018 Coefficient of x^n in (1 + x + n*x^2)^n.

%I A187018 #46 Sep 08 2022 08:45:56
%S A187018 1,1,5,19,145,851,7741,58605,600769,5420035,61026901,628076153,
%T A187018 7648488145,87388647373,1138801242125,14182492489651,196218339243777,
%U A187018 2628971539313875,38377805385510181,547815690902283225,8395817775835635601,126725586542235932329
%N A187018 Coefficient of x^n in (1 + x + n*x^2)^n.
%H A187018 Seiichi Manyama, <a href="/A187018/b187018.txt">Table of n, a(n) for n = 0..590</a> (terms 0..122 from Vincenzo Librandi)
%F A187018 a(n) = [x^n] (1 + x + n*x^2)^n.
%F A187018 a(n) = n^(n/2)*GegenbauerPoly(n,-n,-1/(2*sqrt(n))). - _Emanuele Munarini_, Oct 20 2016
%F A187018 a(n) = Sum_{k=0..floor(n/2)} binomial(n, k)*binomial(n-k, n-2*k)*n^k.
%F A187018 a(n) ~ 2^(n-1/2) * exp(sqrt(n)/2-1/8) * n^(n/2-1/2) / sqrt(Pi). - _Vaclav Kotesovec_, Apr 17 2014
%F A187018 a(n) = [x^n] 1/sqrt(1 - 2*x - (4*n-1)*x^2). - _Paul D. Hanna_, Dec 12 2014
%F A187018 a(n) = n! * [x^n] exp(x) * BesselI(0,2*sqrt(n)*x). - _Ilya Gutkovskiy_, May 31 2020
%e A187018 G.f. = 1 + x + 5*x^2 + 19*x^3 + 145*x^4 + 851*x^5 + 7741*x^6 + 58605*x^7 + ...
%p A187018 A187018:= n -> simplify( n^(n/2)*GegenbauerC(n, -n, -1/(2*sqrt(n))) );
%p A187018 1, seq(A187018(n), n = 1..30); # _G. C. Greubel_, May 31 2020
%t A187018 Flatten[{1,Table[Sum[Binomial[n, k]*Binomial[n-k, n-2*k]*n^k, {k,0,Floor[n/2]}],{n,1,20}]}] (* _Vaclav Kotesovec_, Apr 17 2014 *)
%t A187018 a[ n_]:= SeriesCoefficient[ (1 + x + n*x^2)^n, {x, 0, n}]; (* _Michael Somos_, Dec 12 2014 *)
%t A187018 Table[If[n == 0, 1, Simplify[n^(n/2) GegenbauerC[n, -n, -1/(2 Sqrt[n])]]], {n, 0, 12}] (* _Emanuele Munarini_, Oct 20 2016 *)
%o A187018 (Maxima) a(n):=coeff(expand((1+x+n*x^2)^n),x,n);
%o A187018 makelist(a(n),n,0,20);
%o A187018 (Magma) P<x>:=PolynomialRing(Integers()); [ Coefficients((1+x+n*x^2)^n)[n+1]: n in [0..22] ]; // _Klaus Brockhaus_, Mar 03 2011
%o A187018 (PARI) {a(n)=polcoeff(1/sqrt(1 - 2*x - (4*n-1)*x^2 +x*O(x^n)),n)}
%o A187018 for(n=0,25,print1(a(n),", ")) \\ _Paul D. Hanna_, Dec 12 2014
%o A187018 (PARI) {a(n) = polcoef((1+x+n*x^2)^n, n)} \\ _Seiichi Manyama_, May 01 2019
%o A187018 (Sage) [1]+[ n^(n/2)*gegenbauer(n, -n, -1/(2*sqrt(n))) for n in (1..30)] # _G. C. Greubel_, May 31 2020
%Y A187018 Cf. A092366, A186925, A187019, A187021.
%K A187018 nonn,easy
%O A187018 0,3
%A A187018 _Emanuele Munarini_, Mar 02 2011