A222059 a(n) = n-th harmonic-exponential number, multiplied by n!.

0, 1, 5, 44, 590, 11094, 276924, 8821056, 347992560, 16608856176, 941180477760, 62356907861280, 4768658639919360, 416372600735314560, 41123273761815517440, 4557176483095745510400, 562635159090115071744000, 76906191809174747446425600, 11573912988161070649533849600

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Ayhan Dil and Veli Kurt, Polynomials related to harmonic numbers and evaluation of harmonic number series I, INTEGERS, 12 (2012), #A38. See Eq. (60).

Formula

a(n) = Sum_{k=0..n} A008277(n,k) * (A001008(k)/A002805(k)) * A000142(n). - Michel Marcus, Feb 08 2013

Sum_{n>=0} a(n) * x^n / n!^2 = Sum_{n>=1} H(n) * (exp(x) - 1)^n / n!, where H(n) is the n-th harmonic number. - Ilya Gutkovskiy, Jun 03 2022

Mathematica

Table[Sum[HarmonicNumber[k] StirlingS2[n, k] n!, {k, 0, n}], {n, 0, 20}] (* Vladimir Reshetnikov, Oct 20 2015 *)

Prog

(PARI) a(n) = sum(k=0, n, (sum(i=0, k, (-1)^i*binomial(k, i)*i^n) * (-1)^k/k!)*sum(i=1, k, 1/i) * n!); \\ Michel Marcus, Feb 08 2013

Crossrefs

Cf. A001008, A002805, A008277, A222058.

Sequence in context: A052803 A377860 A201923 * A336290 A214396 A252931

Adjacent sequences: A222056 A222057 A222058 * A222060 A222061 A222062

Keyword

nonn

Author

N. J. A. Sloane, Feb 08 2013

Extensions

More terms from Michel Marcus, Feb 08 2013

Status

approved