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Coefficient of x^n in the expansion of ( 1/(1-x) * (1+x^2)^2 )^n.
2

%I #9 Feb 13 2024 07:37:39

%S 1,1,7,28,143,701,3580,18376,95471,499231,2626607,13883904,73681316,

%T 392323868,2094932728,11214085328,60157698287,323325959395,

%U 1740682221829,9385343934124,50671846382743,273913020523933,1482311190765896,8029798017622048,43538300361416708

%N Coefficient of x^n in the expansion of ( 1/(1-x) * (1+x^2)^2 )^n.

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

%F The g.f. exp( Sum_{k>=1} a(k) * x^k/k ) has integer coefficients and equals (1/x) * Series_Reversion( x * (1-x) / (1+x^2)^2 ). See A369226.

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

%Y Cf. A240688, A370244.

%Y Cf. A369226.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 13 2024