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 A357834 #14 Feb 16 2025 08:34:04
%S A357834 1,0,0,1,-6,35,-224,1603,-12810,113589,-1109472,11852841,-137611110,
%T A357834 1726238787,-23277264192,335861699355,-5164348236138,84316474011861,
%U A357834 -1456893047937600,26562992204112273,-509679388313669574,10266675502780006947,-216625348636705401120
%N A357834 a(n) = Sum_{k=0..floor(n/3)} Stirling1(n,3*k).
%H A357834 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PochhammerSymbol.html">Pochhammer Symbol</a>.
%F A357834 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(log(1+x)).
%F A357834 a(n) = (-1)^n * ( (-1)_n + (-w)_n + (-w^2)_n )/3, where (x)_n is the Pochhammer symbol.
%o A357834 (PARI) a(n) = sum(k=0, n\3, stirling(n, 3*k, 1));
%o A357834 (PARI) my(N=30, x='x+O('x^N)); Vec(serlaplace(sum(k=0, N\3, log(1+x)^(3*k)/(3*k)!)))
%o A357834 (PARI) Pochhammer(x, n) = prod(k=0, n-1, x+k);
%o A357834 a(n) = my(w=(-1+sqrt(3)*I)/2); (-1)^n*round(Pochhammer(-1, n)+Pochhammer(-w, n)+Pochhammer(-w^2, n))/3;
%Y A357834 Cf. A357835, A357836.
%Y A357834 Cf. A105752.
%K A357834 sign
%O A357834 0,5
%A A357834 _Seiichi Manyama_, Oct 14 2022