A317747 Numerator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

1, -1, 1, 259193, -1036793, -201551328007, 9137074752049, 9142431862033871923, -11105299580705049589, -11003865617473929216508154207, 114467620015003245418244743007, 32505236416490926096399421788847363, -254505521478572052318535393350091231, -1828472168539763642032546635313363411876021

Offset

0,4

Comments

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_2 is the second Bendersky constant.

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

Links

Seiichi Manyama, Table of n, a(n) for n = 0..174

Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Formula

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

c_0 = 1, c_n = (2/n) * Sum_{k=0..n-1} B_{n-k+3}*c_k/((n-j+1)*(n-k+2)*(n-k+3)) for n > 0.

a(n) is the numerator of c_n.

Example

1^(1^2)*2^(2^2)*...*n^(n^2) ~ A_2*n^(n^3/3+n^2/2+n/6)*exp(-n^3/9+n/12)*(1 - 1/(360*n) + 1/(259200*n^2) + 259193/(1959552000*n^3) - 1036793/(2821754880000*n^4) - 201551328007/(5079158784000000*n^5) + ... ).

Crossrefs

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2).

Cf. A051675, A243262 (A_2).

Sequence in context: A252921 A216204 A153980 * A252445 A184781 A146897

Adjacent sequences: A317744 A317745 A317746 * A317748 A317749 A317750

Keyword

sign,frac

Author

Seiichi Manyama, Sep 01 2018

Status

approved