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a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.
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%I #12 Oct 18 2024 10:49:20

%S 1,6,27,140,651,2772,11354,44640,169371,624742,2248575,7922124,

%T 27397937,93214632,312559200,1034507696,3384194616,10954244952,

%U 35118346760,111602517096,351819819414,1100912299156,3421515852834,10566654790176,32441857824859,99060134392422

%N a(n) = Sum_{k=0..n} binomial(k+5,5) * binomial(k,n-k)^2.

%F G.f.: (Sum_{k=0..2} A089627(5,k) * (1-x-x^2)^(5-2*k) * x^(3*k)) / ((1-x-x^2)^2 - 4*x^3)^(11/2).

%o (PARI) a(n) = sum(k=0, n, binomial(k+5, 5)*binomial(k, n-k)^2);

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

%o my(N=5, 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, A377148, A377152, A377158, A377159.

%Y Cf. A001874, A089627.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Oct 18 2024