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Coefficient of x^n in the expansion of ( (1-x) / (1-x-x^3) )^n.
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%I #9 May 01 2024 08:58:34

%S 1,0,0,3,4,5,27,63,116,354,945,2123,5563,14846,36519,93083,244068,

%T 622013,1590318,4131265,10658969,27440808,71127683,184324461,

%U 476969939,1237420755,3213687698,8343223779,21682184311,56400917786,146742491187,381991981659

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

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

%o (PARI) a(n, s=3, t=1, u=1) = 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. A054514.

%K nonn

%O 0,4

%A _Seiichi Manyama_, May 01 2024