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

%I #10 Feb 13 2024 07:44:54

%S 1,1,3,16,67,276,1200,5293,23427,104425,468428,2110725,9546256,

%T 43315546,197088195,898910916,4108495491,18812770011,86285313327,

%U 396332663094,1822878714492,8394131895424,38696042930251,178561943852670,824720550229584,3812313399877776

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(2*n-3*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) / (1+x^3)^2 ). See A369266.

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

%Y Cf. A369266, A370214.

%K nonn

%O 0,3

%A _Seiichi Manyama_, Feb 13 2024