A380261 Expansion of e.g.f. exp( ((1+3*x)^(2/3) - 1)/2 ).
1, 1, 0, 2, -14, 146, -1944, 31620, -608068, 13502076, -340052704, 9579145016, -298455813160, 10191129869272, -378469678855904, 15187759126892976, -654936026064200944, 30203464484648818960, -1483333523694819075328, 77291514214052885054496
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Programs
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PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(((1+3*x)^(2/3)-1)/2)))
Formula
a(n) = Sum_{k=0..n} 3^(n-k) * Stirling1(n,k) * A004211(k) = Sum_{k=0..n} 2^k * 3^(n-k) * Stirling1(n,k) * Bell_k(1/2), where Bell_n(x) is n-th Bell polynomial.
a(n) = (1/exp(1/2)) * 3^n * n! * Sum_{k>=0} binomial(2*k/3,n)/(2^k * k!).