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Coefficient of x^n in the expansion of ( (1+x+x^3)^3 / (1+x) )^n.
2

%I #9 Apr 29 2024 09:29:02

%S 1,2,6,29,154,792,4047,20967,110058,582257,3096516,16539690,88684291,

%T 477080671,2573652045,13917256254,75417513498,409447753305,

%U 2226602481387,12126317466294,66129124080804,361059693982863,1973513103986606,10797777359289435

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(3*n,k) * binomial(2*n-k,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) / (1+x+x^3)^3 ). See A372376.

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

%Y Cf. A372373, A372374.

%Y Cf. A372376.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 29 2024