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 A378045 #13 Feb 16 2025 08:34:07
%S A378045 1,2,9,100,1693,39046,1140589,40379872,1680490361,80409242314,
%T A378045 4349556199441,262478904794140,17482853419143061,1274026039224276430,
%U A378045 100830973069183104245,8612770277501109271576,789749958006001265241073,77375794118912255978104978,8066966112797470401673208089
%N A378045 E.g.f. satisfies A(x) = (1+x) * exp(x * A(x)^2 / (1+x)).
%H A378045 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A378045 E.g.f.: (1+x) * exp( -LambertW(-2*x*(1+x))/2 ).
%F A378045 a(n) = n! * Sum_{k=0..n} (2*k+1)^(k-1) * binomial(k+1,n-k)/k!.
%F A378045 a(n) ~ sqrt(1 + 2*exp(-1) - sqrt(1 + 2*exp(-1))) * (1 + sqrt(1 + 2*exp(-1))) * 2^(n-2) * n^(n-1) / ((sqrt(1 + 2*exp(-1)) - 1)^n * exp(n-1)). - _Vaclav Kotesovec_, Nov 15 2024
%o A378045 (PARI) a(n) = n!*sum(k=0, n, (2*k+1)^(k-1)*binomial(k+1, n-k)/k!);
%Y A378045 Cf. A377826, A378046.
%Y A378045 Cf. A377894, A378043.
%Y A378045 Cf. A362773.
%K A378045 nonn
%O A378045 0,2
%A A378045 _Seiichi Manyama_, Nov 15 2024