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

%I #9 Apr 29 2024 09:29:17

%S 1,1,1,7,25,61,187,666,2137,6676,22001,73217,239923,789517,2624182,

%T 8729527,29026553,96790606,323546416,1082566763,3626148425,

%U 12163438539,40847087821,137294721676,461890741843,1555264438186,5240857508017,17672768973979,59634361740734

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(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)^2 ). See A372375.

%o (PARI) a(n, s=3, t=2, u=-1) = sum(k=0, n\s, binomial(t*n, k)*binomial((t+u)*n-k, n-s*k));

%Y Cf. A370185, A370186, A370187.

%Y Cf. A372375.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Apr 28 2024