A370620 Coefficient of x^n in the expansion of 1 / (1-x-x^2)^(3*n).
1, 3, 27, 255, 2535, 25908, 269667, 2843214, 30264975, 324543495, 3500669172, 37940361660, 412830243735, 4507040972190, 49345845670470, 541602648192480, 5957253066586815, 65650003858745514, 724693081872783375, 8011727857439155500, 88692087094226151300
Offset: 0
Keywords
Programs
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Mathematica
a[n_]:=SeriesCoefficient[(1-x-x^2)^(-3*n),{x,0,n}]; Array[a,21,0] (* Stefano Spezia, May 01 2024 *) -
PARI
a(n, s=2, t=3, u=0) = sum(k=0, n\s, binomial(t*n+k-1, k)*binomial((t-u+1)*n-(s-1)*k-1, n-s*k));
Formula
a(n) = Sum_{k=0..floor(n/2)} binomial(3*n+k-1,k) * binomial(4*n-k-1,n-2*k).
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^2)^3 ). See A368963.