The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 56th year, we are closing in on 350,000 sequences, and we’ve crossed 9,700 citations (which often say “discovered thanks to the OEIS”).

 Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A318713 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3). 3
 1, -1, 1513, -127057907, 7078687551763, -1626209947417109183, 25620826938516570309695021, -67861652779316417663427293866727, 11129902336987204608540488473560076627, -2992048697379116617363098289271338606184087563, 593799837691907572156765292649932318031816367209421 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 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). 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. 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. EXAMPLE 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) + ... ). 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). Sequence in context: A282253 A317477 A064584 * A252508 A031810 A020415 Adjacent sequences:  A318710 A318711 A318712 * A318714 A318715 A318716 KEYWORD sign,frac AUTHOR Seiichi Manyama, Sep 01 2018 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified December 5 08:25 EST 2021. Contains 349543 sequences. (Running on oeis4.)