A379186 G.f. A(x) satisfies A(x) = 1/((1 - x*A(x)^3) * (1 - x*A(x))^2).
1, 3, 21, 202, 2270, 27903, 363412, 4927840, 68834941, 983680783, 14312988289, 211329419670, 3158263216267, 47682769300288, 726188701482730, 11142842570134264, 172101193009427174, 2673445730846829604, 41742159037922167264, 654721526817143247304, 10311337739352708700427
Offset: 0
Keywords
Programs
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Mathematica
terms = 21; A[] = 0; Do[A[x] = 1/((1-x*A[x]^3)*(1 -x*A[x])^2) + O[x]^terms // Normal, terms]; CoefficientList[A[x], x] (* Stefano Spezia, Jun 14 2025 *)
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PARI
a(n) = sum(k=0, n, binomial(n+3*k+1, k)*binomial(3*n+3*k+1, n-k)/(n+3*k+1));
Formula
a(n) = Sum_{k=0..n} binomial(n+3*k+1,k) * binomial(3*n+3*k+1,n-k)/(n+3*k+1).