A293318 a(n) = (2*n)! * [x^(2*n)] (-log(sqrt(1 - 2*x)))^n/(sqrt(1 - 2*x)*n!).

1, 4, 86, 3480, 208054, 16486680, 1628301884, 192666441968, 26569595376038, 4184718381424152, 741138328282003860, 145795774074768177360, 31540994233548116475196, 7442380580681963411363440, 1902155375416975061879918520, 523496081998297020687019596000

Offset: 0

Author

Ilya Gutkovskiy, Oct 06 2017

Keywords

• nonn

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..300

Crossrefs

Central terms of triangles A028338, A039757 (gives absolute values) and A109692.

Cf. A265846.

Programs

Mathematica

Table[(2 n)! SeriesCoefficient[(-Log[Sqrt[1 - 2 x]])^n/(Sqrt[1 - 2 x] n!), {x, 0, 2 n}], {n, 0, 15}]

Formula

a(n) ~ c * d^n * (n-1)!, where d = -16*LambertW(-1, -exp(-1/2)/2)^2 / (1 + 2*LambertW(-1, -exp(-1/2)/2)) = 19.643259858273023595... (see also A265846) and c = 1/(2*Pi*sqrt(1 + 1/LambertW(-1, -exp(-1/2)/2))) = 0.2425219128152359859... - Vaclav Kotesovec, Oct 18 2017, updated Mar 17 2024 and May 14 2025