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 A385425 #21 Jun 28 2025 10:01:50
%S A385425 1,1,3,15,113,1145,14499,220703,3932865,80342577,1851286755,
%T A385425 47510525007,1344106404849,41562628517865,1394711974335939,
%U A385425 50480840239135455,1960392617938419969,81309789407316485217,3587373056789171999811,167762667997938465311247
%N A385425 Expansion of e.g.f. exp( -LambertW(-arcsinh(x)) ).
%H A385425 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/LambertW-Function.html">Lambert W-Function</a>.
%F A385425 E.g.f. A(x) satisfies A(x) = exp( arcsinh(x) * A(x) ).
%F A385425 E.g.f. A(x) satisfies A(x) = ( x + sqrt(x^2 + 1) )^A(x).
%F A385425 a(n) = Sum_{k=0..n} (k+1)^(k-1) * i^(n-k) * A385343(n,k), where i is the imaginary unit.
%F A385425 From _Vaclav Kotesovec_, Jun 28 2025: (Start)
%F A385425 a(n) ~ 2^n * exp((exp(-1) - 1)*n + 3/2) * n^(n-1) / (sqrt(1 + exp(2*exp(-1))) * (exp(2*exp(-1)) - 1)^(n - 1/2)).
%F A385425 Equivalently, a(n) ~ n^(n-1) / (sqrt(cosh(exp(-1))) * sinh(exp(-1))^(n - 1/2) * exp(n - 3/2)). (End)
%o A385425 (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(exp(-lambertw(-asinh(x)))))
%Y A385425 Cf. A001147, A385428.
%Y A385425 Cf. A219503, A385343, A385369, A385424.
%K A385425 nonn
%O A385425 0,3
%A A385425 _Seiichi Manyama_, Jun 28 2025