OFFSET
0,2
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..300
FORMULA
a(n) = ((2*n)!/n!) * [x^(2*n)] (-log(1 + log(1 - x)))^n.
From Vaclav Kotesovec, Feb 15 2021: (Start)
a(n) ~ c * d^n * (n-1)!, where
d = -16*p*q^2 * log(-2*q/(1+r))^(1+r) / ((1 + 2*q + r)^2 * (1 + 1/(p*(1+r)))^r) = 17.84101281316291323354184111891200669611476053165484517795417711039479218...
p = LambertW(-1, -1/(exp(1/(1+r))*(1+r)))
q = LambertW(-1, -(1+r)/exp((1+r)/2)/2)
r = 0.5094050884976689299791685259225203723646676600942448390861428232759777841...
is the root of the equation (1+p)*(1+r)^2 * (1 + 2*q + r) * log(-p*(1+r)) + 2*log(-(1+r)/(2*q)) * ((1+q)*(1 + p + p*r) - (1+r) * log(-p*(1+r)) * (p - q + r + p*r + (1+p) * (1+q) * (1+r) * (log(1 + 1/(p*(1+r))) - log(-log(-(1+r)/(2*q)))))) = 0
and c = 0.1417076025518808268972093339771762801784527709... (End)
MATHEMATICA
Table[Sum[Abs[StirlingS1[2 n, k] StirlingS1[k, n]], {k, n, 2 n}], {n, 0, 16}]
Table[((2 n)!/n!) SeriesCoefficient[(-Log[1 + Log[1 - x]])^n, {x, 0, 2 n}], {n, 0, 16}]
PROG
(PARI) a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1)*stirling(k, n, 1))); \\ Michel Marcus, Feb 16 2021
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 15 2021
STATUS
approved