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Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).
2

%I #11 Oct 20 2024 08:32:53

%S 1,0,0,6,6,6,36,66,96,266,576,1026,2246,4866,9516,19598,41286,83526,

%T 170048,351378,716850,1458098,2984028,6087270,12380900,25224222,

%U 51356400,104380510,212164362,431148222,875353220,1776567762,3604752672,7310374010,14819370480,30033014994

%N Expansion of 1/(1 - 4*x^3/(1-x))^(3/2).

%F a(n) = (2*(n-1)*a(n-1) - (n-2)*a(n-2) + 2*(2*n+3)*a(n-3) - 2*(2*n-2)*a(n-4))/n for n > 3.

%F a(n) = Sum_{k=0..floor(n/2)} (2*k+1) * binomial(2*k,k) * binomial(n-2*k-1,n-3*k).

%o (PARI) a(n) = sum(k=0, n\3, (2*k+1)*binomial(2*k, k)*binomial(n-2*k-1, n-3*k));

%Y Cf. A377197, A377204.

%Y Cf. A360309, A377216.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Oct 20 2024