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

0,3

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.

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

Mathematica

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

Crossrefs

Row sums of A222057 or A222060.

Cf. A000254.

Sequence in context: A121124 A180399 A349534 * A265174 A087761 A245503

Adjacent sequences: A222055 A222056 A222057 * A222059 A222060 A222061

Keyword

nonn

Author

N. J. A. Sloane, Feb 08 2013

Status

approved