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 A357831 #15 Feb 16 2025 08:34:04
%S A357831 1,0,0,2,12,70,454,3332,27552,254400,2598852,29125932,355455468,
%T A357831 4693396656,66671326176,1013916648840,16436063079552,282920894841096,
%U A357831 5153797995148296,99052313167391760,2003040751641857856,42513854724369719136,944959706480298199824
%N A357831 a(n) = Sum_{k=0..floor(n/3)} 2^k * |Stirling1(n,3*k)|.
%H A357831 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357831 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) * log(1-x)).
%F A357831 a(n) = ( (2^(1/3))_n + (2^(1/3)*w)_n + (2^(1/3)*w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%o A357831 (PARI) a(n) = sum(k=0, n\3, 2^k*abs(stirling(n, 3*k, 1)));
%o A357831 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k)/(3*k)!)))
%o A357831 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357831 a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round(Pochhammer(v, n)+Pochhammer(v*w, n)+Pochhammer(v*w^2, n))/3;
%Y A357831 Cf. A357832, A357833.
%K A357831 nonn
%O A357831 0,4
%A A357831 _Seiichi Manyama_, Oct 14 2022