cp's OEIS Frontend

A357783 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * Stirling2(n,3*k+1).

%I A357783 #22 Feb 16 2025 08:34:04
%S A357783 0,1,1,1,3,21,131,705,3515,17389,91739,547889,3746227,28241373,
%T A357783 224124083,1821051233,15023818091,126366334125,1094358852075,
%U A357783 9858890038513,92983173940419,918408372280477,9454438841355395,100728532687727329,1103649166937235259
%N A357783 a(n) = Sum_{k=0..floor((n-1)/3)} 2^k * Stirling2(n,3*k+1).
%H A357783 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357783 Let A(0)=1, B(0)=0 and C(0)=0. Let B(n+1) = Sum_{k=0..n} binomial(n,k)*A(k), C(n+1) = Sum_{k=0..n} binomial(n,k)*B(k) and A(n+1) = 2 * Sum_{k=0..n} binomial(n,k)*C(k). A357782(n) = A(n), a(n) = B(n) and A357784(n) = C(n).
%F A357783 Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w^2*exp(w*x) + w*exp(w^2*x))/3 = x + x^4/4! + x^7/7! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1))/(2^(1/3)).
%F A357783 a(n) = ( Bell_n(2^(1/3)) + w^2 * Bell_n(2^(1/3)*w) + w * Bell_n(2^(1/3)*w^2) )/(3*2^(1/3)), where Bell_n(x) is n-th Bell polynomial.
%o A357783 (PARI) a(n) = sum(k=0, (n-1)\3, 2^k*stirling(n, 3*k+1, 2));
%o A357783 (PARI) my(N=30, x='x+O('x^N)); concat(0, Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k+1)/(3*k+1)!))))
%o A357783 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357783 a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Bell_poly(n, v)+w^2*Bell_poly(n, v*w)+w*Bell_poly(n, v*w^2))/(3*v));
%Y A357783 Cf. A357782, A357784.
%Y A357783 Cf. A143816, A357832.
%K A357783 nonn
%O A357783 0,5
%A A357783 _Seiichi Manyama_, Oct 13 2022