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A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.

%I A104631 #28 Apr 04 2025 15:38:18
%S A104631 0,1,4,18,80,365,1686,7875,37080,175725,837100,4004770,19227924,
%T A104631 92599533,447118140,2163837030,10492874384,50972030189,248000853348,
%U A104631 1208335275170,5894873067200,28791371852145,140768761906190
%N A104631 Coefficient of x^(2n+1) in the expansion of (1+x+x^2+x^3+x^4)^n.
%C A104631 In the triangle of pentanomial coefficients, these numbers are in the column next to the central pentanomial coefficients, A005191. Note that for n>0, n divides a(n). This divisibility property is also true for the triangle of trinomial coefficients, A027907, but apparently for no other triangle of m-nomial coefficients. The quotient a(n)/n is in A104632.
%H A104631 Vincenzo Librandi, <a href="/A104631/b104631.txt">Table of n, a(n) for n = 0..300</a>
%H A104631 Andrei G. Pronko, <a href="https://arxiv.org/abs/2504.00835">Periodic Motzkin chain: Ground states and symmetries</a>, arXiv:2504.00835 [math-ph], 2025. See p. 16.
%F A104631 G.f.: sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))). - _Mark van Hoeij_, Nov 16 2011
%F A104631 From _Vaclav Kotesovec_, Oct 17 2012: (Start)
%F A104631 Recurrence: 2*(n-1)*(2*n+1)*a(n) = (19*n^2 - 19*n + 2)*a(n-1) + 5*(2*n^2 - 3*n - 1)*a(n-2) - 25*(n-2)*n*a(n-3).
%F A104631 a(n) ~ 5^n/(2*sqrt(Pi*n)). (End)
%F A104631 a(n) = n * A104632(n) for n>=0. - _Michael Somos_, Aug 12 2018
%e A104631 G.f. = x + 4*x^2 + 18*x^3 + 80*x^4 + 365*x^5 + 1686*x^6 + 7875*x^7 + ... - _Michael Somos_, Aug 12 2018
%t A104631 f=1; Table[f=Expand[f(x^4+x^3+x^2+x+1)]; Coefficient[f, x, 2n+1], {n, 30}]
%o A104631 (Magma) P<x>:=PolynomialRing(Integers()); [n eq 0 select 0 else Coefficients((1+x+x^2+x^3+x^4)^n)[2*n+2]: n in [0..22]]; // _Bruno Berselli_, Nov 17 2011
%o A104631 (PARI) x='x+O('x^30); concat([0], Vec(sqrt((5*x^2+2*x-1+(x+1)*sqrt(5*x^2-6*x+1))/(2*x*(1-x)*(5*x+4)*(5*x-1))))) \\ _G. C. Greubel_, Aug 12 2018
%Y A104631 Cf. A035343 (triangle of pentanomial coefficients).
%Y A104631 Cf. A104632.
%K A104631 nonn,easy
%O A104631 0,3
%A A104631 _T. D. Noe_, Mar 17 2005