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Coefficient of x^n in the expansion of ( (1-x)^2 / (1-x-x^2)^3 )^n.
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%I #15 May 02 2024 09:49:18

%S 1,1,9,46,293,1806,11538,74173,482157,3154645,20762014,137270376,

%T 911111522,6067104434,40514133081,271195540971,1819188150365,

%U 12225956834430,82301499780885,554850642658483,3745615502285478,25315915432984852,171292993893095996

%N Coefficient of x^n in the expansion of ( (1-x)^2 / (1-x-x^2)^3 )^n.

%F a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(2*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)^3 / (1-x)^2 ). See A369488.

%t a[n_]:=SeriesCoefficient[((1-x)^2/(1-x-x^2)^3)^n,{x,0,n}]; Array[a,23,0] (* _Stefano Spezia_, May 01 2024 *)

%o (PARI) a(n, s=2, t=3, u=2) = 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. A370620, A370621, A370623.

%Y Cf. A369488.

%K nonn

%O 0,3

%A _Seiichi Manyama_, May 01 2024