A380257 Expansion of e.g.f. exp( (1/(1-3*x)^(2/3) - 1)/2 ).
1, 1, 6, 56, 706, 11186, 213156, 4742256, 120571676, 3447128796, 109427729096, 3818008773536, 145196289453656, 5976489668054296, 264685744187399536, 12548508890339297856, 634022724191046592016, 34007862777419093053456, 1929842567333195106456416
Offset: 0
Keywords
Links
- Eric Weisstein's World of Mathematics, Bell Polynomial.
Programs
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Mathematica
CoefficientList[Series[Exp[ (1/(1-3*x)^(2/3) - 1)/2 ],{x,0,18}],x]Range[0,18]! (* Stefano Spezia, Mar 31 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(exp((1/(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!).