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

%I #9 Apr 29 2024 09:28:53

%S 1,3,15,93,627,4368,30957,222015,1606803,11713260,85884660,632726988,

%T 4679850525,34729198260,258460942671,1928258130018,14416834880067,

%U 107993546227875,810319315565760,6089298408130707,45821042464807512,345217506895648767

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(3*n,k) * binomial(3*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+x^3)^3 ). See A372378.

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

%Y Cf. A372372, A372373.

%Y Cf. A372378.

%K nonn

%O 0,2

%A _Seiichi Manyama_, Apr 29 2024