login
a(n) = (2*n)! * [x^(2*n)] (-log(sqrt(1 - 2*x)))^n/(sqrt(1 - 2*x)*n!).
5

%I #15 Mar 17 2024 05:35:41

%S 1,4,86,3480,208054,16486680,1628301884,192666441968,26569595376038,

%T 4184718381424152,741138328282003860,145795774074768177360,

%U 31540994233548116475196,7442380580681963411363440,1902155375416975061879918520,523496081998297020687019596000

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

%H Seiichi Manyama, <a href="/A293318/b293318.txt">Table of n, a(n) for n = 0..300</a>

%F 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 = 0.2425219128152359859... - _Vaclav Kotesovec_, Oct 18 2017, updated Mar 17 2024

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

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

%Y Cf. A265846.

%K nonn

%O 0,2

%A _Ilya Gutkovskiy_, Oct 06 2017