A191661 Third differences of A000219.

-1, 3, 0, 9, 1, 22, 12, 48, 45, 120, 125, 290, 354, 676, 913, 1611, 2232, 3757, 5349, 8597, 12462, 19476, 28325, 43445, 63328, 95462, 139139, 207171, 301022, 443779, 642650, 939014, 1354671, 1964715, 2822084, 4066480, 5815907, 8330621, 11863720, 16902592, 23968714, 33981168, 47988828, 67722579, 95258824, 133854462, 187554809, 262483024, 366425586

Offset: 0

Author

N. J. A. Sloane, Jun 10 2011

Keywords

• sign

Comments

From Vaclav Kotesovec, Oct 30 2016: (Start)

More generally, for fixed m > 0, if a(m,n) are m-fold differences of A000219, then

a(m,n) ~ A000219(n) * (2*Zeta[3]/n)^(m/3).

a(m,n) ~ Zeta(3)^(7/36 + m/3) * exp(3 * Zeta(3)^(1/3) * (n/2)^(2/3) + 1/12) / (A * sqrt(3*Pi) * 2^(11/36 - m/3) * n^(25/36 + m/3)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant.

(End)

References

• G. Almkvist, The differences of the number of plane partitions, Manuscript, circa 1991.

Crossrefs

Cf. A000219, A191659, A191660.

Programs

Mathematica

nmax = 50; Drop[CoefficientList[Series[(1-x)^3 * Product[1/(1-x^k)^k, {k, 1, nmax}], {x, 0, nmax}], x], 3] (* Vaclav Kotesovec, Oct 30 2016 *)

Formula

a(n) ~ 2^(25/36) * Zeta(3)^(43/36) * exp(1/12 + 3*Zeta(3)^(1/3)*n^(2/3)/2^(2/3)) / (A * sqrt(3*Pi) * n^(61/36)), where Zeta(3) = A002117 and A = A074962 is the Glaisher-Kinkelin constant. - Vaclav Kotesovec, Oct 30 2016