A318713 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3).

1, -1, 1513, -127057907, 7078687551763, -1626209947417109183, 25620826938516570309695021, -67861652779316417663427293866727, 11129902336987204608540488473560076627, -2992048697379116617363098289271338606184087563, 593799837691907572156765292649932318031816367209421

Offset: 0

Author

Seiichi Manyama, Sep 01 2018

Keywords

• sign
• frac

Comments

1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(Sum_{k>=0} b(k)/n^k)^n, where A_3 is the third Bendersky constant.

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

Examples

			1^(1^3)*2^(2^3)*...*n^(n^3) ~ A_3*n^(n^4/4+n^3/2+n^2/4-1/120)*exp(-n^4/16+n^/12)*(1 - 1/(5040*n^2) + 1513/(50803200*n^4) - 127057907/(8449588224000*n^6) + 7078687551763/(442893616349184000*n^8) - 1626209947417109183/(55804595659997184000000*n^10) + ... ).
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..106
• Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

Crossrefs

Product_{z=1..n} z^(z^m): A001163/A001164 (m=0), A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).

Cf. A243263 (A_3).

Formula

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

c_0 = 1, c_n = (-3/n) * Sum_{k=0..n-1} B_{2*n-2*k+4}*c_k/((2*n-2*k+1)*(2*n-2*k+2)*(2*n-2*k+3)*(2*n-2*k+4)) for n > 0.

a(n) is the numerator of c_n.