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Coefficient of x^n in the expansion of ( (1-x+x^3) / (1-x) )^(2*n).
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%I #10 Apr 30 2024 05:31:55

%S 1,0,0,6,8,10,78,196,376,1446,4390,10648,32822,101426,276976,808666,

%T 2449528,7046942,20491458,61124482,179376718,525065722,1556298700,

%U 4598892274,13546834582,40109057710,118836735758,351539306142,1041872654824,3091535558296

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

%F a(n) = Sum_{k=0..floor(n/3)} binomial(2*n,k) * binomial(n-2*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)^2 / (1-x+x^3)^2 ). See A372418.

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

%Y Cf. A372417, A372418.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Apr 29 2024