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

%I #10 May 01 2024 08:59:27

%S 1,2,14,98,726,5522,42770,335512,2656998,21195944,170076214,

%T 1371181110,11098310730,90128497032,734008622872,5992486341248,

%U 49028047353670,401885885751630,3299812135410080,27134786911366212,223433635272820126,1842041118321640390

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

%F a(n) = Sum_{k=0..floor(n/2)} binomial(2*n+k-1,k) * binomial(3*n-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-x^2)^2 ). See A368961.

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

%Y Cf. A370618, A370619.

%Y Cf. A368961.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 30 2024