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A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.

%I A377152 #20 Dec 06 2024 01:54:50
%S A377152 1,5,20,95,400,1561,5915,21610,76585,265075,898622,2992235,9810290,
%T A377152 31727815,101379175,320464280,1003259080,3113576320,9586763720,
%U A377152 29305985800,88997753446,268642069750,806394498200,2408144329250,7157177344225,21177323087891
%N A377152 a(n) = Sum_{k=0..n} binomial(k+4,4) * binomial(k,n-k)^2.
%H A377152 Robert Israel, <a href="/A377152/b377152.txt">Table of n, a(n) for n = 0..2357</a>
%F A377152 G.f.: (Sum_{k=0..2} A089627(4,k) * (1-x-x^2)^(4-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(9/2).
%p A377152 f:= proc(n) local k; add(binomial(k+4,4)*binomial(k,n-k)^2,k=0..n) end proc:
%p A377152 map(f, [$0..50]); # _Robert Israel_, Dec 05 2024
%o A377152 (PARI) a(n) = sum(k=0, n, binomial(k+4, 4)*binomial(k, n-k)^2);
%o A377152 (PARI) a089627(n, k) = n!/((n-2*k)!*k!^2);
%o A377152 my(N=4, M=30, x='x+O('x^M), X=1-x-x^2, Y=3); Vec(sum(k=0, N\2, a089627(N, k)*X^(N-2*k)*x^(Y*k))/(X^2-4*x^Y)^(N+1/2))
%Y A377152 Cf. A051286, A182884, A377145, A377148, A377153, A377158, A377159.
%Y A377152 Cf. A001873, A089627.
%K A377152 nonn
%O A377152 0,2
%A A377152 _Seiichi Manyama_, Oct 18 2024