A370622 Coefficient of x^n in the expansion of ( (1-x)^2 / (1-x-x^2)^3 )^n.
1, 1, 9, 46, 293, 1806, 11538, 74173, 482157, 3154645, 20762014, 137270376, 911111522, 6067104434, 40514133081, 271195540971, 1819188150365, 12225956834430, 82301499780885, 554850642658483, 3745615502285478, 25315915432984852, 171292993893095996
Offset: 0
Keywords
Programs
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Mathematica
a[n_]:=SeriesCoefficient[((1-x)^2/(1-x-x^2)^3)^n,{x,0,n}]; Array[a,23,0] (* Stefano Spezia, May 01 2024 *) -
PARI
a(n, s=2, t=3, u=2) = 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(2*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 / (1-x)^2 ). See A369488.