A143475 -id:A143475 - cp's OEIS frontend

A001164 Stirling's formula: denominators of asymptotic series for Gamma function.

1, 12, 288, 51840, 2488320, 209018880, 75246796800, 902961561600, 86684309913600, 514904800886784000, 86504006548979712000, 13494625021640835072000, 9716130015581401251840000, 116593560186976815022080000, 2798245444487443560529920000, 299692087104605205332754432000000, 57540880724084199423888850944000000

Offset: 0

N. J. A. Sloane

			Gamma(z) ~ sqrt(2 Pi) z^(z-1/2) e^(-z) (1 + 1/(12 z) + 1/(288 z^2) - 139/(51840 z^3) - 571/(2488320 z^4) + ... ), z -> infinity in |arg z| < Pi.

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapters 2 and 43, equations 2:6:1 and 43:6:6 at pages 21, 415.

Seiichi Manyama, Table of n, a(n) for n = 0..295 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
S. Brassesco and M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894 [math.CA], 2010.
V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 (2009), 839-843.
Peter Luschny, Approximations to the factorial function.
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829. MR1080390 (92b:41049)
T. Müller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
Richard M. Slevinsky, On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev-Jacobi transform, arXiv preprint arXiv:1602.02618 [math.NA], 2016.
N. M. Temme, The asymptotic expansion of the incomplete gamma function, SIAM J. Math. Anal., 10 (1979), 757-766. [From _N. J. A. Sloane_, Feb 20 2012]
Nico Temme, Uniform Asymptotics for the incomplete gamma functions starting from negative values of the parameters, arXiv:math/9603218 [math.CA], 1996.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Eric Weisstein's World of Mathematics, Stirlings Series
J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.

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

h := proc(k) option remember; local j; `if`(k=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1),j=1..k-1))/(1+1/(k+1))) end:
coeffStirling := n -> h(2*n)*doublefactorial(2*n-1):
seq(denom(coeffStirling(n)), n=0..16); # Peter Luschny, Nov 05 2015

Denominator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
h[k_] := h[k] = If[k==0, 1, (h[k-1]/k-Sum[h[k-j]*h[j]/(j+1), {j, 1, k-1}]) / (1+1/(k+1))]; StirlingAsympt[n_] := h[2n]*2^n*Pochhammer[1/2, n]; a[n_] := StirlingAsympt[n] // Denominator; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 12 2015, after Peter Luschny *)

a(n)=local(A,m); if(n<1,n==0,A=vector(m=2*n+1,k,1); for(k=2,m,A[k]=(A[k-1]-sum(i=2,k-1,i*A[i]*A[k+1-i]))/(k+1)); denominator(A[m]*m!/2^n/n!)) /* Michael Somos, Jun 09 2004 */

The coefficients c_k have g.f. 1 + Sum_{k >= 1} c_k/z^k = exp( Sum_{k >= 1} B_{2k}/(2k(2k-1)z^(2k-1)) ).
Numerators/denominators: A001163(n)/A001164(n) = (6*n+1)!!/(4^n*(2*n)!) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i)* C(i,j)*C(j,k)/((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients. - Vladimir Reshetnikov, Nov 04 2015
a(n) = denominator(h(2*n)*doublefactorial(2*n-1)) where h(k) = (h(k-1)/k - Sum_{j=1..k-1} h(k-j)*h(j)/(j+1))/(1+1/(k+1)) and h(0)=1. - Peter Luschny, Nov 05 2015
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-1} B_{n-k+1}*c_k/(n-k+1) for n > 0. Then a(n) is the denominator of c_n. - Seiichi Manyama, Sep 01 2018

More terms from Vladeta Jovovic, Nov 14 2001

A001163 Stirling's formula: numerators of asymptotic series for Gamma function.

Original entry on oeis.org

1, 1, 1, -139, -571, 163879, 5246819, -534703531, -4483131259, 432261921612371, 6232523202521089, -25834629665134204969, -1579029138854919086429, 746590869962651602203151, 1511513601028097903631961, -8849272268392873147705987190261, -142801712490607530608130701097701

Offset: 0

N. J. A. Sloane

			Gamma(z) ~ sqrt(2*Pi) * z^(z-1/2) * e^(-z) * (1 + 1/(12*z) + 1/(288*z^2) - 139/(51840*z^3) - 571/(2488320*z^4) + ... ), z -> infinity in |arg z| < Pi.

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 267, #23.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
Jerome Spanier and Keith B. Oldham, "Atlas of Functions", Hemisphere Publishing Corp., 1987, chapter 2, equation 2:6:1 at page 21.

Seiichi Manyama, Table of n, a(n) for n = 0..227 (terms 0..100 from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math.Series 55, Tenth Printing, 1972, p. 257, Eq. 6.1.37.
S. Brassesco and M. A. Méndez, The asymptotic expansion for the factorial and Lagrange inversion formula, arXiv:1002.3894 [math.CA], 2010.
V. De Angelis, Stirling's series revisited, Amer. Math. Monthly, 116 (2009), 839-843.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1
Peter Luschny, Approximations to the factorial function.
G. Marsaglia and J. C. W. Marsaglia, A new derivation of Stirling's approximation to n!, Amer. Math. Monthly, 97 (1990), 827-829. MR1080390 (92b:41049)
T. Mueller, Finite group actions and asymptotic expansion of e^P(z), Combinatorica, 17 (4) (1997), 523-554.
Richard M. Slevinsky, On the use of Hahn's asymptotic formula and stabilized recurrence for a fast, simple, and stable Chebyshev-Jacobi transform, arXiv preprint arXiv:1602.02618 [math.NA], 2016.
N. M. Temme, The asymptotic expansion of the incomplete gamma function, SIAM J. Math. Anal., 10 (1979), 757-766. [From _N. J. A. Sloane_, Feb 20 2012]
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).
Eric Weisstein's World of Mathematics, Stirling's Series.
J. W. Wrench, Jr., Concerning two series for the gamma function, Math. Comp., 22 (1968), 617-626.

Cf. A001164 (denominators).
Cf. A097303 (see W. Lang link there for a similar numerator sequence which deviates for the first time at entry number 33. Expansion of GAMMA(z) in terms of 1/(k!*z^k) instead of 1/z^k).
Product_{z=1..n} z^(z^m): A143475/A143476 (m=1), A317747/A317796 (m=2), A318713/A318714 (m=3).

h := proc(k) option remember; local j; `if`(k=0, 1,
(h(k-1)/k-add((h(k-j)*h(j))/(j+1), j=1..k-1))/(1+1/(k+1))) end:
StirlingAsympt := proc(n) option remember; h(2*n)*2^n*pochhammer(1/2, n) end:
A001163 := n -> numer(StirlingAsympt(n));
seq(A001163(n), n=0..30); # Peter Luschny, Feb 08 2011

Numerator[ Reverse[ Drop[ CoefficientList[ Simplify[ PowerExpand[ Normal[ Series[n!, {n, Infinity, 17}]]Exp[n]/(Sqrt[2Pi n]*n^(n - 17))]], n], 1]]]
(* Second program: *)
h[k_] := h[k] = If[k==0, 1, (h[k-1]/k-Sum[h[k-j]*h[j]/(j+1), {j, 1, k-1}]) / (1+1/(k+1))];
StirlingAsympt[n_] := h[2n]*2^n*Pochhammer[1/2, n];
a[n_] := StirlingAsympt[n] // Numerator;
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Oct 12 2015, after Peter Luschny *)

a(n)=local(A,m); if(n<1,n==0,A=vector(m=2*n+1,k,1); for(k=2,m,A[k]=(A[k-1]-sum(i=2,k-1,i*A[i]*A[k+1-i]))/(k+1)); numerator(A[m]*m!/2^n/n!)) /* Michael Somos, Jun 09 2004 */

def A001163(n):
    @cached_function
    def h(k):
        if k<=0: return 1
        S = sum((h(k-j)*h(j))/(j+1) for j in (1..k-1))
        return (h(k-1)/k-S)/(1+1/(k+1))
    return numerator(h(2*n)*2^n*rising_factorial(1/2,n))
[A001163(n) for n in range(17)] # Peter Luschny, Nov 05 2015

The coefficients c_k have g.f. 1 + Sum_{k >= 1} c_k/z^k = exp( Sum_{k >= 1} B_{2k}/(2k(2k-1)z^(2k-1)) ).
Numerators/denominators: A001163(n)/A001164(n) = (6*n+1)!!/(4^n*(2*n)!) * Sum_{i=0..2*n} Sum_{j=0..i} Sum_{k=0..j} (-1)^k*2^i*k^(2*n+i+j)*C(2*n,i) *C(i,j)*C(j,k)/((2*n+2*i+1)*(2*n+i+j)!), assuming 0^0 = 1 (when n = 0), n!! = A006882(n), C(n,k) = A007318(n,k) are binomial coefficients. - Vladimir Reshetnikov, Nov 05 2015
From Seiichi Manyama, Sep 01 2018: (Start)
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-1} B_{n-k+1}*c_k/(n-k+1) for n > 0.
a(n) is the numerator of c_n. (End)

More terms from Vladeta Jovovic, Nov 14 2001

Signs added by Robert G. Wilson v, Jul 12 2003

A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

Original entry on oeis.org

1, 720, 7257600, 15676416000, 3476402012160000, 162695614169088000000, 4919915372473221120000000, 60219764159072226508800000000, 507464726196802564122476544000000000, 3288371425755280615513648005120000000000

Offset: 0

Eric W. Weisstein, Aug 19 2008

In Glaisher (1878) equation (2) is "1^1.2^2.3^3 ... n^n = A n^(n^2/2 + n/2 + 1/12) e^(-n^4/4) (1 + 1/(720n^2) - 1433/(7257600n^4) + &c.)" - Michael Somos, Jun 24 2012

			(Glaisher*(1 - 1433/(7257600*z^4) + 1/(720*z^2))*z^(1/12 + (z*(1 + z))/2))/e^(z^2/4).
From _Seiichi Manyama_, Aug 31 2018: (Start)
c_1 = -1/2 * (B_4*c_0/(3*4)) = 1/720, so a(1) = 720.
c_2 = -1/4 * (B_6*c_0/(5*6) + B_4*c_1/(3*4)) = -1433/7257600, so a(2) = 7257600. (End)

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
J. W. L. Glaisher, On The Product 1^1.2^2.3^3 ... n^n, Messenger of Mathematics, 7 (1878), pp. 43-47, see p. 43 eq. (2)

Seiichi Manyama, Table of n, a(n) for n = 0..148
Jean-Christophe Pain, Series representations for the logarithm of the Glaisher-Kinkelin constant, arXiv:2304.07629 [math.NT], 2023.
Eric Weisstein's World of Mathematics, Hyperfactorial

Cf. A002109, A143475.

From Seiichi Manyama, Aug 31 2018: (Start)
Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (-1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
a(n) is the denominator of c_n. (End)

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Sep 01 2018

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).

			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) + ... ).

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.

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

Cf. A051675, A243262 (A_2).

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.

A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

Original entry on oeis.org

1, 360, 259200, 1959552000, 2821754880000, 5079158784000000, 76796880814080000000, 304115648023756800000000, 125121866615488512000000000, 258236518070374430146560000000000, 929651465053347948527616000000000000, 334674527419205261469941760000000000000, 920050700832433373350094438400000000000000

Offset: 0

Seiichi Manyama, Sep 01 2018

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 denominator of b(n).

			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) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..206
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

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

Cf. A051675, A243262 (A_2).

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 denominator of c_n.

A317615 Numerator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).

Original entry on oeis.org

1, 0, 0, 1, 0, -1, 1, 1, -1, -2591989, 1, 143285617, -576460261, -56447987, 3300470359, 3143536325702272141, -1786064113, -75094364111707432631, 2817261842693900500163, 669559278432109042489, -328057861344412119208697, -10210033960347221028684507558893, 4858439165868890899345523

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 numerator 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..266
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. A143475, A317660.

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 numerator of c_n.

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

Original entry on oeis.org

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

Offset: 0

Seiichi Manyama, Sep 01 2018

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).

			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) + ... ).

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.

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).

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.

A318714 Denominator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^3).

Original entry on oeis.org

1, 5040, 50803200, 8449588224000, 442893616349184000, 55804595659997184000000, 315568291905804875857920000000, 211531737430299124385080934400000000, 6522145617145034649275530739712000000000, 254485460571619683408716971558739902464000000000

Offset: 0

Seiichi Manyama, Sep 01 2018

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).

			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) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..134
Weiping Wang, Some asymptotic expansions on hyperfactorial functions and generalized Glaisher-Kinkelin constants, ResearchGate, 2017.

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).

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.

A318710 Numerator of the coefficient of z^(-2*n) in the Stirling-like asymptotic expansion of G(z+1), where G(z) is Barnes G-function.

Original entry on oeis.org

1, -1, 1447, -1559527, 366331136219, -637231027521743, 2629597771763437160249, -9781318441276304057417323, 5699253125605574587097648227233017, -13391188869589008440145241321782451523, 33214021675956829606886933935672301973543264421

Offset: 0

Seiichi Manyama, Sep 01 2018

G(z+1) ~ A^(-1)*z^(-z^2/2-z/2-1/12)*exp(z^2/4)*(Gamma(z+1))^z*(Sum_{n>=0} b(n)/z^(2*n)), where A is the Glaisher-Kinkelin constant and Gamma is the gamma function.

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

			G(z+1) ~ A^(-1)*z^(-z^2/2-z/2-1/12)*exp(z^2/4)*(Gamma(z+1))^z*(1 - 1/(720*z^2) + 1447/(7257600*z^4) - 1559527/(15676416000*z^6) + 366331136219/(3476402012160000*z^8) - 637231027521743/(3320318656512000000*z^10) + ... ).

Seiichi Manyama, Table of n, a(n) for n = 0..114
Chao-Ping Chen, Asymptotic expansions for Barnes G-function, Journal of Number Theory 135 (2014) 36-42.
Eric Weisstein's World of Mathematics, Barnes G-Function

Cf. A143475, A318711.

Let B_n be the Bernoulli number, and define the sequence {c_n} by the recurrence
c_0 = 1, c_n = (1/(2*n)) * Sum_{k=0..n-1} B_{2*n-2*k+2}*c_k/((2*n-2*k+1)*(2*n-2*k+2)) for n > 0.
a(n) is the numerator of c_n.

cp's OEIS Frontend

A001164 Stirling's formula: denominators of asymptotic series for Gamma function.

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A001163 Stirling's formula: numerators of asymptotic series for Gamma function.

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A143476 Denominator of the coefficient of z^(2n) in the Stirling-like asymptotic expansion of the hyperfactorial function A002109.

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

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A317796 Denominator of the coefficient of z^(-n) in the Stirling-like asymptotic expansion of Product_{z=1..n} z^(z^2).

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A317615 Numerator of the coefficient of z^(-n) of asymptotic expansions related to hyperfactorial function H(z).

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

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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).

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