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 A349719 #27 Feb 16 2025 08:34:02
%S A349719 1,1,0,1,-4,26,-212,2108,-24720,334072,-5112544,87396728,-1650607040,
%T A349719 34132685120,-767025716736,18612106195456,-485013257865472,
%U A349719 13509071081429888,-400505695457942528,12592502771190979712,-418524228123134068224
%N A349719 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x))/2 ).
%H A349719 Seiichi Manyama, <a href="/A349719/b349719.txt">Table of n, a(n) for n = 0..411</a>
%H A349719 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349719 a(n) = (1/2^n) * Sum_{k=0..n} (-k+1)^(n-1) * binomial(n,k).
%F A349719 E.g.f.: (x/2)/LambertW( x/2 * exp(-x/2) ).
%F A349719 G.f.: 2 * Sum_{k>=0} (-k+1)^(k-1) * x^k/(2 - (-k+1)*x)^(k+1).
%F A349719 a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n-1)). - _Vaclav Kotesovec_, Dec 05 2021
%t A349719 a[n_] := (1/2^n) * Sum[If[k == n == 1, 1, (-k + 1)^(n - 1)] * Binomial[n, k], {k, 0, n}]; Array[a, 21, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349719 (PARI) a(n) = sum(k=0, n, (-k+1)^(n-1)*binomial(n, k))/2^n;
%o A349719 (PARI) my(N=40, x='x+O('x^N)); Vec(serlaplace((x/2)/lambertw(x/2*exp(-x/2))))
%o A349719 (PARI) my(N=40, x='x+O('x^N)); Vec(2*sum(k=0, N, (-k+1)^(k-1)*x^k/(2-(-k+1)*x)^(k+1)))
%Y A349719 Cf. A007889, A202617, A283828, A349714, A349715, A349716, A349720, A349721.
%K A349719 sign
%O A349719 0,5
%A A349719 _Seiichi Manyama_, Nov 27 2021