A143475 Numerator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

1, 1, -1433, 1550887, -365236274341, 31170363588856607, -2626723351027654662151, 127061942835077684151157039, -5696145248370283185291966600124423, 254326794362835881966596504823903633657, -33203124408022060010631772664020406983485604379

Offset

0,3

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.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..113

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

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 numerator of c_n. (End)

Example

(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) = 1.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = -1433. (End)

Crossrefs

Cf. A002109, A143476.

Sequence in context: A205070 A169822 A114083 * A245948 A221004 A204862

Adjacent sequences: A143472 A143473 A143474 * A143476 A143477 A143478

Keyword

sign,frac

Author

Eric W. Weisstein, Aug 19 2008

Extensions

More terms from Seiichi Manyama, Aug 31 2018

Status

approved