A317660 Denominator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).

1, 1, 1, 720, 1, 5040, 1036800, 10080, 3628800, 24634368000, 6350400, 747242496000, 3476402012160000, 105670656000, 11298306539520000, 1489290622009344000000, 2259661307904000, 6688268793387417600000, 920024174652492349440000000, 8655406673795481600000

Offset

0,4

Comments

1^1*2^2*...*n^n ~ A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4)*(Sum_{k>=0} b(k)/n^k)^n, where A is the Glaisher-Kinkelin constant.

a(n) is the denominator of b(n).

Links

Seiichi Manyama, Table of n, a(n) for n = 0..362

Chao-Ping Chen, Asymptotic expansions for Barnes G-function, Journal of Number Theory 135 (2014) 36-42.

Eric Weisstein's World of Mathematics, Hyperfactorial

Formula

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

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

a(n) is the denominator of c_n.

Example

1^1*2^2*...*n^n ~ A*n^(n^2/2 + n/2 + 1/12)*exp(-n^2/4)*(1 + 1/(720*n^3) - 1/(5040*n^5) + 1/(1036800*n^6) + 1/(10080*n^7) - 1/(3628800*n^8) - 2591989/(24634368000*n^9) + ... )^n.

Crossrefs

Cf. A143476, A317615.

Sequence in context: A267336 A167984 A267429 * A267571 A267287 A267746

Adjacent sequences: A317657 A317658 A317659 * A317661 A317662 A317663

Keyword

nonn,frac

Author

Seiichi Manyama, Sep 01 2018

Status

approved