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%I #25 Oct 13 2018 13:42:13
%S 0,1,4,21,138,1095,10208,109473,1328470,18003675,269580492,4420677525,
%T 78801184322,1517300654415,31386251780536,694190761402377,
%U 16348768018619694,408472183061464515,10791720442056792740,300605598797790229629,8805117712245004098586,270562051319419652165175,8702576800277309526639504,292425620801795849417200881
%N Harmonic-geometric numbers.
%H Vincenzo Librandi, <a href="/A222058/b222058.txt">Table of n, a(n) for n = 0..200</a>
%H Ayhan Dil and Veli Kurt, <a href="https://www.emis.de/journals/INTEGERS/papers/m38/m38.Abstract.html">Polynomials related to harmonic numbers and evaluation of harmonic number series I</a>, INTEGERS, 12 (2012), #A38.
%F a(n) = Sum_{k=0..n} Stirling2(n,k)*|Stirling1(k+1,2)|.
%F Maximal term in the sum is asymptotically in position k = n/(2*log(2)) and limit n-> infinity (a(n)/n!)^(1/n) = 1/log(2). - _Vaclav Kotesovec_, Feb 09 2013
%F E.g.f.: -log(2 - exp(x))/(2 - exp(x)). - _Ilya Gutkovskiy_, May 31 2018
%F a(n) ~ n! * log(n) / (2 * (log(2))^(n+1)) * (1 + (gamma - log(2) - log(log(2))) / log(n)), where gamma is the Euler-Mascheroni constant A001620. - _Vaclav Kotesovec_, Oct 13 2018
%t Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[k + 1, 2]], {k, 0, n}], {n, 0, 20}] (* _Vaclav Kotesovec_, Feb 09 2013 *)
%Y Row sums of A222057 or A222060.
%Y Cf. A000254.
%K nonn
%O 0,3
%A _N. J. A. Sloane_, Feb 08 2013
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