cp's OEIS Frontend

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

%I A341589 #12 Feb 17 2021 01:57:42
%S A341589 1,2,40,1485,81088,5856900,526685269,56704848200,7112345477952,
%T A341589 1018548226480356,163987811350464660,29321558852248050388,
%U A341589 5764958268855541178967,1236150756215397667568170,287086392921014590422630300,71789589754855255636302048525,19231403740347427723119910379040
%N A341589 a(n) = Sum_{k=n..2*n} |Stirling1(2*n, k) * Stirling1(k, n)|.
%H A341589 Vaclav Kotesovec, <a href="/A341589/b341589.txt">Table of n, a(n) for n = 0..300</a>
%F A341589 a(n) = ((2*n)!/n!) * [x^(2*n)] (-log(1 + log(1 - x)))^n.
%F A341589 From _Vaclav Kotesovec_, Feb 15 2021: (Start)
%F A341589 a(n) ~ c * d^n * (n-1)!, where
%F A341589 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...
%F A341589 p = LambertW(-1, -1/(exp(1/(1+r))*(1+r)))
%F A341589 q = LambertW(-1, -(1+r)/exp((1+r)/2)/2)
%F A341589 r = 0.5094050884976689299791685259225203723646676600942448390861428232759777841...
%F A341589 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
%F A341589 and c = 0.1417076025518808268972093339771762801784527709... (End)
%t A341589 Table[Sum[Abs[StirlingS1[2 n, k] StirlingS1[k, n]], {k, n, 2 n}], {n, 0, 16}]
%t A341589 Table[((2 n)!/n!) SeriesCoefficient[(-Log[1 + Log[1 - x]])^n, {x, 0, 2 n}], {n, 0, 16}]
%o A341589 (PARI) a(n) = sum(k=n, 2*n, abs(stirling(2*n, k, 1)*stirling(k, n, 1))); \\ _Michel Marcus_, Feb 16 2021
%Y A341589 Cf. A003713, A008275, A039814, A321712, A341587, A341588.
%K A341589 nonn
%O A341589 0,2
%A A341589 _Ilya Gutkovskiy_, Feb 15 2021