A352075 Expansion of e.g.f. sqrt(1 - log(1 - 2*x)).
1, 1, 1, 5, 25, 209, 1961, 23589, 321105, 5100801, 90384369, 1792247973, 39011436201, 928869511569, 23953711043289, 666047439187077, 19847286284835105, 631267636613496705, 21339849019758468705, 764149215124570567365, 28891697037933017586105
Offset: 0
Keywords
Programs
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Mathematica
m = 20; Range[0, m]! * CoefficientList[Series[(1 - Log[1 - 2*x])^(1/2), {x, 0, m}], x] (* Amiram Eldar, Mar 05 2022 *) -
PARI
my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(1-log(1-2*x)))) -
PARI
a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, -2*j+1)*stirling(n, k, 1));
Formula
a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (-2*j+1)) * Stirling1(n,k).
a(n) ~ n! * 2^(n-1) / (sqrt(log(n)) * n) * (1 - (gamma + 1)/(2*log(n))), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Mar 05 2022