A222058 Harmonic-geometric numbers.

0, 1, 4, 21, 138, 1095, 10208, 109473, 1328470, 18003675, 269580492, 4420677525, 78801184322, 1517300654415, 31386251780536, 694190761402377, 16348768018619694, 408472183061464515, 10791720442056792740, 300605598797790229629, 8805117712245004098586, 270562051319419652165175, 8702576800277309526639504, 292425620801795849417200881

Offset: 0

Author

N. J. A. Sloane, Feb 08 2013

Keywords

• nonn

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38.

Crossrefs

Row sums of A222057 or A222060.

Cf. A000254.

Programs

Mathematica

Table[Sum[StirlingS2[n, k]*Abs[StirlingS1[k + 1, 2]], {k, 0, n}], {n, 0, 20}] (* Vaclav Kotesovec, Feb 09 2013 *)

Formula

a(n) = Sum_{k=0..n} Stirling2(n,k)*|Stirling1(k+1,2)|.

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

E.g.f.: -log(2 - exp(x))/(2 - exp(x)). - Ilya Gutkovskiy, May 31 2018

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