A372110 G.f. A(x) satisfies A(x) = ( (1 - x*A(x))/(1 - 10*x*A(x)) )^(1/3).
1, 3, 30, 381, 5457, 84000, 1356726, 22680705, 389100000, 6811276449, 121177168266, 2184600000000, 39822674320065, 732762138176436, 13592289000000000, 253896500477864361, 4771765283550516435, 90167361600000000000, 1712019315455953465026
Offset: 0
Keywords
Programs
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PARI
a(n) = sum(k=0, n, 9^k*binomial(n/3+k-2/3, k)*binomial(n-1, n-k))/(n+1);
Formula
a(n) = (1/(n+1)) * Sum_{k=0..n} 9^k * binomial(n/3+k-2/3,k) * binomial(n-1,n-k).
From Seiichi Manyama, Nov 30 2024: (Start)
G.f.: exp( Sum_{k>=1} A378552(k) * x^k/k ).
a(n) = (1/(n+1)) * [x^n] 1/(1 - 9*x/(1-x))^((n+1)/3).
G.f.: (1/x) * Series_Reversion( x*(1 - 9*x/(1-x))^(1/3) ). (End)