A317615 -id:A317615 - cp's OEIS frontend

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

Seiichi Manyama, Sep 01 2018

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).

			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.

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

Cf. A143476, A317615.

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.

A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

Original entry on oeis.org

1, 6, 256, 28560, 6071040, 2098483200, 1071889920000, 758870167910400, 711206089850880000, 852336059876720640000, 1271438437097485762560000, 2310211006286602237378560000, 5023141810386294125321256960000, 12877606625796048169971744768000000, 38439740210093310755176533983232000000

Offset: 0

Stefano Spezia, Apr 30 2023

nonn

Christian Krattenthaler and Thomas W. Müller, The congruence properties of Romik's sequence of Taylor coefficients of Jacobi's theta function theta_3, arXiv:2304.11471 [math.NT], 2023. See p. 5.

Cf. A317615, A362714, A362715.

Table[(2n+1)!SeriesCoefficient[x*Hypergeometric2F1[3/4, 3/4, 3/2, 4*x^2]/Hypergeometric2F1[1/4, 1/4, 1/2, 4*x^2], {x, 0, 2n+1}], {n,0, 14}]
(* or *)
a[0]=1; a[n_]:=Product[(4j-1)^2,{j,n}]-Sum[Binomial[2n+1,2m+1]Product[(4j-3)^2,{j,n-m}]a[m],{m,0,n-1}]; Array[a,15,0]

a(n) = Product_{j=1..n} (4*j - 1)^2 - Sum_{m=0..n-1} binomial(2*n+1, 2*m+1)*Product_{j=1..n-m} (4*j - 3)^2*a(m) for n > 0.

cp's OEIS Frontend

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

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A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

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cp's OEIS Frontend

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A362713 Expansion of e.g.f. x*2F1([3/4, 3/4], [3/2], 4*x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.

Views

Author

Keywords

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Programs

Mathematica

Formula

A362713 Expansion of e.g.f. x2F1([3/4, 3/4], [3/2], 4x^2)/2F1([1/4, 1/4], [1/2], 4*x^2), odd powers only.