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

%I #9 Apr 29 2024 09:28:58

%S 1,1,1,10,37,91,334,1366,4645,15967,59951,220782,792946,2906554,

%T 10770082,39629440,145966549,540943231,2006762563,7443051014,

%U 27661527427,102980882455,383639407570,1430429881122,5339465251426,19947662875216,74573064834646

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(3*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)^2 / (1+x+x^3)^3 ). See A372377.

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

%Y Cf. A372372, A372374.

%Y Cf. A372377.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Apr 29 2024