A341589 a(n) = Sum_{k=n..2*n} |Stirling1(2*n, k) * Stirling1(k, n)|.

1, 2, 40, 1485, 81088, 5856900, 526685269, 56704848200, 7112345477952, 1018548226480356, 163987811350464660, 29321558852248050388, 5764958268855541178967, 1236150756215397667568170, 287086392921014590422630300, 71789589754855255636302048525, 19231403740347427723119910379040

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

Cf. A003713, A008275, A039814, A321712, A341587, A341588.

Sequence in context: A012834 A304448 A012241 * A286124 A275931 A099707

Adjacent sequences: A341586 A341587 A341588 * A341590 A341591 A341592

Keyword

nonn

Author

Ilya Gutkovskiy, Feb 15 2021

Status

approved