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 A357833 #15 Feb 16 2025 08:34:04
%S A357833 0,0,1,3,11,52,304,2114,16992,154626,1568706,17535108,213965520,
%T A357833 2828584824,40259041188,613656673476,9971942784132,172071391424832,
%U A357833 3141974627361216,60523400730707208,1226519845766281008,26084378634267048984,580854626450078463000
%N A357833 a(n) = Sum_{k=0..floor((n-2)/3)} 2^k * |Stirling1(n,3*k+2)|.
%H A357833 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357833 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) * log(1-x))/(2^(2/3)).
%F A357833 a(n) = ( (2^(1/3))_n + w * (2^(1/3)*w)_n + w^2 * (2^(1/3)*w^2)_n )/(3*2^(2/3)), where (x)_n is the Pochhammer symbol.
%o A357833 (PARI) a(n) = sum(k=0, (n-2)\3, 2^k*abs(stirling(n, 3*k+2, 1)));
%o A357833 (PARI) my(N=30, x='x+O('x^N)); concat([0, 0], Vec(serlaplace(sum(k=0, N\3, 2^k*(-log(1-x))^(3*k+2)/(3*k+2)!))))
%o A357833 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357833 a(n) = my(v=2^(1/3), w=(-1+sqrt(3)*I)/2); round((Pochhammer(v, n)+w*Pochhammer(v*w, n)+w^2*Pochhammer(v*w^2, n))/(3*v^2));
%Y A357833 Cf. A357831, A357832.
%K A357833 nonn
%O A357833 0,4
%A A357833 _Seiichi Manyama_, Oct 14 2022