This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A357782 #24 Feb 16 2025 08:34:04
%S A357782 1,0,0,2,12,50,184,686,2996,16642,110328,784190,5645876,40685762,
%T A357782 296458344,2226254766,17564381332,147289101090,1312394060536,
%U A357782 12305546886398,119906479624084,1202273551045474,12341175064817576,129582557972751918,1394497073432776756
%N A357782 a(n) = Sum_{k=0..floor(n/3)} 2^k * Stirling2(n,3*k).
%H A357782 Eric Weisstein's MathWorld, <a href="https://mathworld.wolfram.com/BellPolynomial.html">Bell Polynomial</a>.
%F A357782 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). a(n) = A(n), A357783(n) = B(n) and A357784(n) = C(n).
%F A357782 Let w = exp(2*Pi*i/3) and set F(x) = (exp(x) + exp(w*x) + exp(w^2*x))/3 = 1 + x^3/3! + x^6/6! + ... . Then the e.g.f. for the sequence is F(2^(1/3) * (exp(x)-1)).
%F A357782 a(n) = ( Bell_n(2^(1/3)) + Bell_n(2^(1/3)*w) + Bell_n(2^(1/3)*w^2) )/3, where Bell_n(x) is n-th Bell polynomial.
%o A357782 (PARI) a(n) = sum(k=0, n\3, 2^k*stirling(n, 3*k, 2));
%o A357782 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(exp(x)-1)^(3*k)/(3*k)!)))
%o A357782 (PARI) Bell_poly(n, x) = exp(-x)*suminf(k=0, k^n*x^k/k!);
%o A357782 a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Bell_poly(n, v)+Bell_poly(n, v*w)+Bell_poly(n, v*w^2))/3;
%Y A357782 Cf. A357783, A357784.
%Y A357782 Cf. A143815, A357831.
%K A357782 nonn
%O A357782 0,4
%A A357782 _Seiichi Manyama_, Oct 13 2022