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 A349528 #31 Feb 16 2025 08:34:02
%S A349528 1,1,6,80,1645,45962,1627080,69817575,3522349232,204343964292,
%T A349528 13403304111515,980876342339456,79235384391436316,7003257362607771709,
%U A349528 672285536392973397658,69656231091367157111844,7747832754070176901631621
%N A349528 a(n) = Sum_{k=0..n} (-1)^(n-k) * (3*k+1)^(k-1) * Stirling2(n,k).
%H A349528 Seiichi Manyama, <a href="/A349528/b349528.txt">Table of n, a(n) for n = 0..336</a>
%H A349528 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349528 E.g.f. satisfies: log(A(x)) = (1 - exp(-x)) * A(x)^3.
%F A349528 E.g.f.: exp( -LambertW(3 * (exp(-x) - 1))/3 ).
%F A349528 G.f.: Sum_{k>=0} (3*k+1)^(k-1) * x^k/Product_{j=1..k} (1 + j*x).
%F A349528 a(n) ~ sqrt(3*exp(1) - 1) * sqrt(log(3/(3-exp(-1)))) * n^(n-1) / (3 * exp(n - 1/3) * (1 + log(3/(3*exp(1) - 1)))^n). - _Vaclav Kotesovec_, Nov 21 2021
%t A349528 a[n_] := Sum[(-1)^(n - k)*(3*k + 1)^(k - 1) * StirlingS2[n, k], {k, 0, n}]; Array[a, 17, 0] (* _Amiram Eldar_, Nov 21 2021 *)
%o A349528 (PARI) a(n) = sum(k=0, n, (-1)^(n-k)*(3*k+1)^(k-1)*stirling(n, k, 2));
%o A349528 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(3*(exp(-x)-1))/3)))
%o A349528 (PARI) my(N=20, x='x+O('x^N)); Vec(sum(k=0, N, (3*k+1)^(k-1)*x^k/prod(j=1, k, 1+j*x)))
%Y A349528 Cf. A008277, A058864, A196556, A349525, A349527.
%K A349528 nonn
%O A349528 0,3
%A A349528 _Seiichi Manyama_, Nov 20 2021