A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

1, 720, 7257600, 15676416000, 3476402012160000, 162695614169088000000, 4919915372473221120000000, 60219764159072226508800000000, 507464726196802564122476544000000000, 3288371425755280615513648005120000000000

Offset: 0

Author

Eric W. Weisstein, Aug 19 2008

Keywords

• nonn
• frac

Comments

In Glaisher (1878) equation (2) is "1^1.2^2.3^3 ... n^n = A n^(n^2/2 + n/2 + 1/12) e^(-n^4/4) (1 + 1/(720n^2) - 1433/(7257600n^4) + &c.)" - Michael Somos, Jun 24 2012

Examples

			(Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
From _Seiichi Manyama_, Aug 31 2018: (Start)
c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 720.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = 7257600. (End)
		

References

• Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
• J. W. L. Glaisher, On The Product 1^1.2^2.3^3 ... n^n, Messenger of Mathematics, 7 (1878), pp. 43-47, see p. 43 eq. (2)

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..148
• Jean-Christophe Pain, Series representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2304.07629 [math.NT], 2023.
• Eric Weisstein's World of Mathematics, Hyperfactorial

Crossrefs

Cf. A002109, A143475.

Formula

From Seiichi Manyama, Aug 31 2018: (Start)

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence

c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.

a(n) is the denominator of c_n. (End)