A352075 - OEIS

%I #29 Mar 06 2022 08:33:51

%S 1,1,1,5,25,209,1961,23589,321105,5100801,90384369,1792247973,

%T 39011436201,928869511569,23953711043289,666047439187077,

%U 19847286284835105,631267636613496705,21339849019758468705,764149215124570567365,28891697037933017586105

%N Expansion of e.g.f. sqrt(1 - log(1 - 2*x)).

%F a(n) = Sum_{k=0..n} (-2)^(n-k) * (Product_{j=0..k-1} (-2*j+1)) * Stirling1(n,k).

%F 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

%t m = 20; Range[0, m]! * CoefficientList[Series[(1 - Log[1 - 2*x])^(1/2), {x, 0, m}], x] (* _Amiram Eldar_, Mar 05 2022 *)

%o (PARI) my(N=20, x='x+O('x^N)); Vec(serlaplace(sqrt(1-log(1-2*x))))

%o (PARI) a(n) = sum(k=0, n, (-2)^(n-k)*prod(j=0, k-1, -2*j+1)*stirling(n, k, 1));

%Y Cf. A352113, A352114.

%Y Cf. A097397.

%K nonn

%O 0,4

%A _Seiichi Manyama_, Mar 05 2022