The On-Line Encyclopedia of Integer Sequences (OEIS)

Revision History for A191661 (Bold, blue-underlined text is an addition; faded, red-underlined text is a ~~deletion~~.)
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A191661

Third differences of A000219.
(history; published version)

#12 by Vaclav Kotesovec at Sun Oct 30 18:57:36 EDT 2016

STATUS

editing

approved

#11 by Vaclav Kotesovec at Sun Oct 30 18:56:40 EDT 2016

COMMENTS

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

#10 by Vaclav Kotesovec at Sun Oct 30 18:46:43 EDT 2016

COMMENTS

From Vaclav Kotesovec, Oct 30 2016: (Start)

More generally, for fixed m > 0, if a(m,n) is 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)

#9 by Vaclav Kotesovec at Sun Oct 30 18:31:00 EDT 2016

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 *)

#8 by Vaclav Kotesovec at Sun Oct 30 18:27:28 EDT 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

STATUS

approved

editing

#7 by Russ Cox at Fri Mar 30 16:52:02 EDT 2012

AUTHOR

_N. J. A. Sloane (njas(AT)research.att.com), _, Jun 10 2011

Discussion

Fri Mar 30

16:52

OEIS Server: https://oeis.org/edit/global/110

#6 by N. J. A. Sloane at Fri Jun 10 10:17:58 EDT 2011

STATUS

proposed

approved

#5 by N. J. A. Sloane at Fri Jun 10 10:17:56 EDT 2011

CROSSREFS

Cf. A000219, A191659, A191660.

STATUS

approved

proposed

#4 by N. J. A. Sloane at Fri Jun 10 10:14:18 EDT 2011

STATUS

proposed

approved

#3 by N. J. A. Sloane at Fri Jun 10 10:14:13 EDT 2011

REFERENCES

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