A385367 Expansion of e.g.f. 1/(1 - 2 * arcsinh(x)).
1, 2, 8, 46, 352, 3378, 38912, 522702, 8024064, 138586722, 2659565568, 56141737518, 1292851544064, 32253357421842, 866534937329664, 24943658876605902, 765883864848531456, 24985882009464388290, 863077992845681885184, 31469256501815056673070
Offset: 0
Programs
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Mathematica
With[{nn=20},CoefficientList[Series[1/(1-2ArcSinh[x]),{x,0,nn}],x] Range[0,nn]!] (* Harvey P. Dale, Jul 14 2025 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(1/(1-2*asinh(x))))
Formula
E.g.f.: 1/(1 - 2 * log(x + sqrt(x^2 + 1))).
E.g.f.: B(x)^2, where B(x) is the e.g.f. of A385371.
a(n) = Sum_{k=0..n} 2^k * k! * i^(n-k) * A385343(n,k), where i is the imaginary unit.
a(n) ~ sqrt(Pi) * (1 + exp(1)) * 2^(n - 1/2) * n^(n + 1/2) / ((exp(1) - 1)^(n+1) * exp(n/2)). - Vaclav Kotesovec, Jun 27 2025