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a(n) = Sum_{k=0..n} binomial(k+3,3) * binomial(k,n-k)^2.
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%I #14 Oct 18 2024 10:49:12

%S 1,4,14,60,225,796,2764,9304,30580,98700,313422,981548,3037473,

%T 9301620,28222000,84927760,253699285,752863840,2220831160,6515581600,

%U 19021079866,55276625304,159967084164,461150383400,1324652146775,3792447499916,10824189204014

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

%F G.f.: (1-x-x^2) * ((1-x-x^2)^2 + 6*x^3) / ((1-x-x^2)^2 - 4*x^3)^(7/2).

%o (PARI) a(n) = sum(k=0, n, binomial(k+3, 3)*binomial(k, n-k)^2);

%o (PARI) a089627(n, k) = n!/((n-2*k)!*k!^2);

%o 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))

%Y Cf. A051286, A182884, A377145, A377152, A377153, A377158, A377159.

%Y Cf. A377149, A377150.

%Y Cf. A089627.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 18 2024