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 A349721 #22 Feb 16 2025 08:34:02
%S A349721 1,1,-2,19,-260,4966,-121328,3613996,-127035920,5147600680,
%T A349721 -236245559984,12112405259560,-686148484748480,42560312499982720,
%U A349721 -2868921992458611200,208828244778853125376,-16324500711130356582656,1363986660232205656646272
%N A349721 E.g.f. satisfies: A(x) = exp( x * (1 + 1/A(x)^3)/2 ).
%H A349721 Seiichi Manyama, <a href="/A349721/b349721.txt">Table of n, a(n) for n = 0..352</a>
%H A349721 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A349721 a(n) = (1/2^n) * Sum_{k=0..n} (-3*k+1)^(n-1) * binomial(n,k).
%F A349721 E.g.f.: ( (3*x/2)/LambertW( 3*x/2 * exp(-3*x/2) ) )^(1/3).
%F A349721 G.f.: 2 * Sum_{k>=0} (-3*k+1)^(k-1) * x^k/(2 - (-3*k+1)*x)^(k+1).
%F A349721 a(n) ~ -(-1)^n * sqrt(1 + LambertW(exp(-1))) * 3^(n-1) * n^(n-1) / (2^n * exp(n) * LambertW(exp(-1))^(n - 1/3)). - _Vaclav Kotesovec_, Dec 05 2021
%t A349721 a[n_] := (1/2^n) * Sum[(-3*k + 1)^(n - 1) * Binomial[n, k], {k, 0, n}]; Array[a, 18, 0] (* _Amiram Eldar_, Nov 27 2021 *)
%o A349721 (PARI) a(n) = sum(k=0, n, (-3*k+1)^(n-1)*binomial(n, k))/2^n;
%o A349721 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(((3*x/2)/lambertw(3*x/2*exp(-3*x/2)))^(1/3)))
%o A349721 (PARI) my(N=20, x='x+O('x^N)); Vec(2*sum(k=0, N, (-3*k+1)^(k-1)*x^k/(2-(-3*k+1)*x)^(k+1)))
%Y A349721 Cf. A007889, A202617, A349714, A349715, A349716, A349719, A349720.
%K A349721 sign
%O A349721 0,3
%A A349721 _Seiichi Manyama_, Nov 27 2021