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A377150 a(n) = Sum_{k=0..floor(n/3)} binomial(k+3,3) * binomial(k,n-3*k)^2.

%I A377150 #16 Dec 06 2024 01:55:11
%S A377150 1,0,0,4,4,0,10,40,10,20,180,180,55,560,1260,616,1435,5600,5684,4424,
%T A377150 18956,33720,24780,55944,147249,157560,182280,523540,826440,802560,
%U A377150 1681966,3531880,4072035,5671084,12941764,19281064,22523175,43823520,80254746,99744776
%N A377150 a(n) = Sum_{k=0..floor(n/3)} binomial(k+3,3) * binomial(k,n-3*k)^2.
%H A377150 Robert Israel, <a href="/A377150/b377150.txt">Table of n, a(n) for n = 0..5705</a>
%F A377150 G.f.: (1-x^3-x^4) * ((1-x^3-x^4)^2 + 6*x^7) / ((1-x^3-x^4)^2 - 4*x^7)^(7/2).
%p A377150 f:= proc(n) local k; add(binomial(k+3,3)*binomial(k,n-3*k)^2,k=0..floor(n/3)) end proc:
%p A377150 map(f, [$0..50]); # _Robert Israel_, Dec 05 2024
%o A377150 (PARI) a(n) = sum(k=0, n\3, binomial(k+3, 3)*binomial(k, n-3*k)^2);
%o A377150 (PARI) a089627(n, k) = n!/((n-2*k)!*k!^2);
%o A377150 my(N=3, M=40, x='x+O('x^M), X=1-x^3-x^4, Y=7); 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 A377150 Cf. A377148, A377149.
%Y A377150 Cf. A089627.
%K A377150 nonn
%O A377150 0,4
%A A377150 _Seiichi Manyama_, Oct 18 2024