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A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.

%I A377148 #19 May 12 2025 11:54:27
%S A377148 1,4,14,60,225,796,2764,9304,30580,98700,313422,981548,3037473,
%T A377148 9301620,28222000,84927760,253699285,752863840,2220831160,6515581600,
%U A377148 19021079866,55276625304,159967084164,461150383400,1324652146775,3792447499916,10824189204014
%N A377148 a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.
%H A377148 Vincenzo Librandi, <a href="/A377148/b377148.txt">Table of n, a(n) for n = 0..1000</a>
%F A377148 G.f.: (1-x-x^2) * ((1-x-x^2)^2 + 6*x^3) / ((1-x-x^2)^2 - 4*x^3)^(7/2).
%t A377148 Table[Sum[Binomial[k+3,3]*Binomial[k, n-k]^2,{k,0,n}],{n,0,30}] (* _Vincenzo Librandi_, May 12 2025 *)
%o A377148 (PARI) a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(k, n-k)^2);
%o A377148 (PARI) a089627(n, k) = n!/((n-2*k)!*k!^2);
%o A377148 my(N=3, 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))
%o A377148 (Magma) [&+[Binomial(k+3,3)*Binomial(k, n-k)^2: k in [0..n]]: n in [0..30]]; // _Vincenzo Librandi_, May 12 2025
%Y A377148 Cf. A051286, A182884, A377145, A377152, A377153, A377158, A377159.
%Y A377148 Cf. A377149, A377150.
%Y A377148 Cf. A089627.
%K A377148 nonn
%O A377148 0,2
%A A377148 _Seiichi Manyama_, Oct 18 2024