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

Author

N. J. A. Sloane, Feb 08 2013

Keywords

• nonn

Links

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

Crossrefs

Cf. A001008, A002805, A008277, A222058.

Programs

Mathematica

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

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

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

Extensions

More terms from Michel Marcus, Feb 08 2013