%I #12 Oct 30 2016 18:57:36

%S -1,3,0,9,1,22,12,48,45,120,125,290,354,676,913,1611,2232,3757,5349,

%T 8597,12462,19476,28325,43445,63328,95462,139139,207171,301022,443779,

%U 642650,939014,1354671,1964715,2822084,4066480,5815907,8330621,11863720,16902592,23968714,33981168,47988828,67722579,95258824,133854462,187554809,262483024,366425586

%N Third differences of A000219.

%C From _Vaclav Kotesovec_, Oct 30 2016: (Start)

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

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

%C 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.

%C (End)

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

%F 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

%t 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 *)

%Y Cf. A000219, A191659, A191660.

%K sign

%O 0,2

%A _N. J. A. Sloane_, Jun 10 2011