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A357784 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * Stirling2(n,3*k+2).

%I A357784 #21 Feb 16 2025 08:34:04
%S A357784 0,0,1,3,7,17,61,343,2231,14301,88561,542011,3397483,22638993,
%T A357784 164336085,1299899087,10991061663,97070035205,881323166809,
%U A357784 8173386231395,77489746906355,754631383660729,7590899551399869,79174328607339767,856889470005396071
%N A357784 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * Stirling2(n,3*k+2).
%H A357784 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357784 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), A357783(n) = B(n) and a(n) = C(n).
%F A357784 Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + w*exp(w*x) + w^2*exp(w^2*x))/3 = x^2/2! + x^5/5! + x^8/8! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1))/(2^(2/3)).
%F A357784 a(n) = ( Bell_n(2^(1/3)) + w * Bell_n(2^(1/3)*w) + w^2 * Bell_n(2^(1/3)*w^2) )/(3*2^(2/3)), where Bell_n(x) is n-th Bell polynomial.
%o A357784 (PARI) a(n) = sum(k=0, (n-2)\3, 2^k*stirling(n, 3*k+2, 2));
%o A357784 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k+2)/(3*k+2)!))))
%o A357784 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357784 a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Bell_poly(n, v)+w*Bell_poly(n, v*w)+w^2*Bell_poly(n, v*w^2))/(3*v^2));
%Y A357784 Cf. A357782, A357783.
%Y A357784 Cf. A143817, A357833.
%K A357784 nonn
%O A357784 0,4
%A A357784 _Seiichi Manyama_, Oct 13 2022