A222060 -id:A222060 - cp's OEIS frontend

A222057 Triangle read by rows: coefficients of harmonic-geometric polynomials.

1, 1, 3, 1, 9, 11, 1, 21, 66, 50, 1, 45, 275, 500, 274, 1, 93, 990, 3250, 4110, 1764, 1, 189, 3311, 17500, 38360, 37044, 13068, 1, 381, 10626, 85050, 287700, 469224, 365904, 109584, 1, 765, 33275, 388500, 1904574, 4667544, 6037416, 3945024, 1026576, 1, 1533, 102630, 1705250, 11651850, 40266828, 76839840, 82188000, 46195920, 10628640

Offset: 1

N. J. A. Sloane, Feb 08 2013

			Triangle begins:
  1;
  1,   3;
  1,   9,    11;
  1,  21,    66,    50;
  1,  45,   275,   500,    274;
  1,  93,   990,  3250,   4110,   1764;
  1, 189,  3311, 17500,  38360,  37044,  13068;
  1, 381, 10626, 85050, 287700, 469224, 365904, 109584;
  ...

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

Row sums give A222058. See A222060 for another version (including row & column 0).

A222057(n,k)=stirling(n,k,2)*abs(stirling(k+1,2)) \\ with 1 <= k <= n: vector(8,n,vector(n,k,A222057(n,k))). - M. F. Hasler, Jul 12 2018

The n-th polynomial is Sum_{k=0..n} Stirling2(n,k)*|Stirling1(k+1,2)|*x^k.

(The k=0 term is always 0. Sequence lists coefficients of x, x^2, x^3, ... - M. F. Hasler, Jul 12 2018)

A222058 Harmonic-geometric numbers.

Original entry on oeis.org

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

N. J. A. Sloane, Feb 08 2013

nonn

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.

Row sums of A222057 or A222060.

Cf. A000254.

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

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

cp's OEIS Frontend

A222057 Triangle read by rows: coefficients of harmonic-geometric polynomials.

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