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A318714 Denominator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3). 3
1, 5040, 50803200, 8449588224000, 442893616349184000, 55804595659997184000000, 315568291905804875857920000000, 211531737430299124385080934400000000, 6522145617145034649275530739712000000000, 254485460571619683408716971558739902464000000000 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,2
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 denominator of b(n).
LINKS
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 denominator 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: A008552 A221437 A221622 * A227669 A010800 A172544
KEYWORD
nonn,frac
AUTHOR
Seiichi Manyama, Sep 01 2018
STATUS
approved

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Last modified March 28 13:25 EDT 2024. Contains 371254 sequences. (Running on oeis4.)