A001164 -id:A001164 - cp's OEIS frontend

A182913 Denominators of an asymptotic series for the Gamma function (G. Nemes).

1, 1, 144, 12960, 207360, 2612736, 9405849600, 18811699200, 1083553873920, 4022693756928000, 300361133850624000, 210853515963138048000, 151814531493459394560000, 151814531493459394560000, 21861292535058152816640000

Offset: 0

Peter Luschny, Feb 09 2011

G_n = A182912(n)/A182913(n). These rational numbers provide the coefficients for an asymptotic expansion of the Gamma function.

			G_0 = 1, G_1 = 0, G_2 = 1/144, G_3 = -1/12960.

G. Nemes, More Accurate Approximations for the Gamma Function,
Thai Journal of Mathematics Volume 9(1) (2011), 21-28.

Peter Luschny, Approximation Formulas for the Factorial Function.

Cf. A001163, A001164, A182912.

# See A182912 for G(n).
A182913 := n -> denom(G(n)); seq(A182913(i),i=0..15);

G[n_] := G[n] = Module[{j, J}, J[k_] := J[k] = Module[{j}, If[k == 0, 1, (J[k-1]/k - Sum[J[k-j]*J[j]/(j+1), {j, 1, k-1}])/(1+1/(k+1))]]; Sum[J[2*j]*2^j*6^(j-n)*Gamma[1/2+j]/(Gamma[n-j+1]*Gamma[1/2+j-n]), {j, 0, n}] - Sum[G[j]*(-4)^(j-n)*Gamma[n]/(Gamma[n-j+1]*Gamma[j]), {j, 1, n-1}]]; A182913[n_] := Denominator[G[n]]; Table[A182913[i], {i, 0, 15}] (* Jean-François Alcover, Jan 06 2014, translated from Maple *)

Gamma(x+1) ~ x^x e^(-x) sqrt(2Pi (x+1/6)) Sum_{n>=0} G_n / (x+1/4)^n.

A182935 Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

Original entry on oeis.org

1, -1, 1, 1003, -4027, -5128423, 168359651, 68168266699, -587283555451, -221322134443186643, 3253248645450176257, 52946591945344238676937, -3276995262387193162157789, -6120218676760621380031990351

Offset: 0

Peter Luschny, Feb 24 2011

G_n = A182935(n)/A144618(n). These rational numbers provide the coefficients for an asymptotic expansion of the factorial function.

The relationship between these coefficients and the Bernoulli numbers are due to De Moivre, 1730 (see Laurie).

			G_0 = 1, G_1 = -1/24, G_2 = 1/1152, G_3 = 1003/414720.

Dirk Laurie, Old and new ways of computing the gamma function, page 14, 2005.
Peter Luschny, Approximation Formulas for the Factorial Function.
W. Wang, Unified approaches to the approximations of the gamma function, J. Number Theory (2016).

Cf. A001163, A001164, A144618.

G := proc(n) option remember; local j,R;
R := seq(2*j,j=1..iquo(n+1,2));
`if`(n=0,1,add(bernoulli(j,1/2)*G(n-j+1)/(n*j),j=R)) end:
A182935 := n -> numer(G(n)); seq(A182935(i),i=0..15);

a[0] = 1; a[n_] := a[n] = Sum[ BernoulliB[j, 1/2]*a[n-j+1]/(n*j), {j, 2, n+1, 2}]; Table[a[n] // Numerator, {n, 0, 15}] (* Jean-François Alcover, Jul 26 2013, after Maple *)

z! ~ sqrt(2 Pi) (z+1/2)^(z+1/2) e^(-z-1/2) Sum_{n>=0} G_n / (z+1/2)^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.

A321938 Denominators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).

Original entry on oeis.org

1, 12, 288, 51840, 2488320, 209018880, 75246796800, 180592312320, 86684309913600, 73557828698112000, 86504006548979712000, 13494625021640835072000, 9716130015581401251840000, 23318712037395363004416000, 559649088897488712105984000

Offset: 0

Richard P. Brent, Nov 22 2018

The Maclaurin coefficients arise in a theorem of Slater (1960) on asymptotic expansions of confluent hypergeometric functions, see Sec. 3.1 of the paper by Temme (2013), and Theorem 5 of the preprint by Brent et al. (2018).

The sequence is related to A001164 but differs from the 7th term.

			For n=0..3 the Maclaurin coefficients are 1, -1/12, 1/288, 67/61840.

L. J. Slater, Confluent Hypergeometric Functions, Cambridge University Press, 1960.

Richard P. Brent, M. L. Glasser, Anthony J. Guttmann, A Conjectured Integer Sequence Arising From the Exponential Integral, arXiv:1812.00316 [math.NT], 2018.
N. M. Temme, Remarks on Slater's asymptotic expansions of Kummer functions for large values of the a-parameter, Adv. Dyn. Syst. Appl., 8 (2013), 365-377.

Numerators are A321937.

A321938List := proc(len) local mu, ser;
mu  := h -> sum(bernoulli(2*k)/(2*k)!*h^(2*k-1), k=1..infinity);
ser := series(exp(mu(h)), h, len+2): seq(denom(coeff(ser,h,n)), n=0..len) end:
A321938List(14); # Peter Luschny, Dec 05 2018

Exp[1/x - 1/(Exp[x]-1) - 1/2] + O[x]^20 // CoefficientList[#, x]& // Denominator (* Jean-François Alcover, Jan 21 2019 *)

x='x+O('x^25); apply(denominator, Vec(exp(1/x - 1/(exp(x)-1) - 1/2)))  \\ Joerg Arndt, Dec 05 2018

A362113 Truncate Stirling's asymptotic series for 1! after n terms and round to the nearest integer.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 0, 12, 13, -131, -144, 1878, 2047, -31243, -34023, 603493, 656720, -13392786, -14565501, 338472513, 367934625, -9665776360, -10502979551, 309738982467, 336455915833, -11068897604205, -12020303454921, 438669580592210

Offset: 0

N. J. A. Sloane, Apr 15 2023

sign

Stirling's series for N! is an asymptotic expansion. It does not converge to N! as more terms are included in the sum.

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)

Cf. A001163/A001164, A362114-A362116.

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:
c := n -> StirlingAsympt(n); # # Peter Luschny, Feb 08 2011 (This is A001163(n)/A001164(n)).
S:=proc(k,N) local i; global c; sqrt(2*Pi)*N^(N+1/2)*exp(-N)*add(c(i)/N^i,i=0..k); end;
Digits:=200;
T:=proc(N,M) local k; [seq(round(evalf(S(k,N))),k=0..M)]; end;
T(1,40);

In general, we take Stirling's asymptotic series for N! (N >= 1, with N = 1 for the present sequence) and truncate it after n terms. This has the value
sqrt(2*Pi)*N^(N+1/2)*exp(-N)*(Sum_{j = 0..n} c(j)/N^j),
where c(j) = A001163(j)/A001164(j).
We then round this to the nearest integer to get a(n).

A362116 Truncate Stirling's asymptotic series for 4! after n terms and round to the nearest integer.

Original entry on oeis.org

24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 24, 23, 23, 28, 28, -6, -7, 271, 276, -2098, -2148, 19280, 19727, -184017, -188262, 1850289, 1892769, -19543114, -19989790, 216612777, 221542994

Offset: 0

N. J. A. Sloane, Apr 15 2023

sign

See A362113 for further information.

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)

Cf. A001163/A001164, A362113-A362115.

More than the usual number of terms are shown in order to demonstrate the divergence of the sequence.

A097301 Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.

Original entry on oeis.org

1, -1, 2, -3, 3360, -995040, 39916800, -656924748480, 1214047650816000, -169382556838010880, 15749593891765493760000, -4054844479616799289344000, 34017686450062663131463680000, -11402327189708082115897599590400000, 189528830020089532044244068728832000000

Offset: 0

Wolfdieter Lang, Aug 13 2004

Denominators are given in A097302.
The e.g.f. sum( A(2*n+1)*(x^(2*n+1))/(2*n+1)!,n=0..infinity) appears in the Stirling-formula derivation for N! with x=1/N in the exponent and the formula for A(2*n+1):=a(n)/A097302(n), n>=0, is given below. For Stirling's formula see A001163 and A001164.
The rationals A(2*n+1) = B(n):= (2*n)!*Bernoulli(2*(n+1))/(2*(n+1)) = a(n)/A097304(n) with A(2*n):=0 are the logarithmic transform of the rational sequence {A001163(n)/A001164(n)} (inverse of the sequence transform EXP)

Julian Havil, Gamma, Exploring Euler's Constant, Princeton University Press, Princeton and Oxford, 2003, p. 87.

W. Lang, More terms and comments.
N. J. A. Sloane, Transforms

a(n)=numerator(B(n)) with B(n):=Bernoulli(2*n+2)*(2*n)!/(2*n+2) and Bernoulli(n)= A027641(n)/A027642(n).

A097303 Denominators in Stirling's asymptotic series.

Original entry on oeis.org

1, 12, 144, 8640, 103680, 1741824, 104509440, 179159040, 2149908480, 1418939596800, 23838185226240, 338068808663040, 20284128519782400, 18723810941337600, 32097961613721600, 229179445921972224000

Offset: 0

Wolfdieter Lang, Aug 13 2004

Numerators coincide with the numbers depicted in A001163 but differ for the first time at entry nr. 33. See the W. Lang link.

Stirling's formula for Gamma(z) (|arg(z)| < Pi) uses the asymptotic series Sum_{k>=0} (N(k)/a(k))*((1/z)^k)/k!. For N(k) see the W. Lang link.

W. Lang, More terms and comments.

Cf. A001163, A001164 (Stirling formula with further links and references.).

max = 15; se = Series[(E^x*Sqrt[1/x]*Gamma[x+1])/(x^x*Sqrt[2*Pi]), {x, Infinity, max}]; Denominator[ CoefficientList[ se /. x -> 1/x, x]*Range[0, max]!] (* Jean-François Alcover, Nov 03 2011 *)

a(n) = denominator(s(n)), where the signed rationals s(n) are the coefficients of ((1/z)^k)/k! in the asymptotic series appearing in Stirling's formula for Gamma(z).

A122252 Binet's factorial series. Numerators of the coefficients of a convergent series for the logarithm of the Gamma function.

Original entry on oeis.org

1, 1, 59, 29, 533, 1577, 280361, 69311, 36226519, 7178335, 64766889203, 32128227179, 459253205417, 325788932161, 2311165698322609, 287144996287039, 1215091897184850539, 402833263943353393, 476099430416027805187, 236881416523193720213, 650730651653461090091101

Offset: 1

Paul Drees (zemyla(AT)gmail.com), Aug 27 2006

			Rational sequence starts: 1/12, 1/12, 59/360, 29/60, 533/280, 1577/168, 280361/5040, ...
c(1) = Integral_{x=0..1} x*(x - 1/2) / 1 = Integral_{x=0..1} (x^2 - x/2) = (x^3/3 - x^2/4) | {x, 0, 1} = 1/12.

Robert G. Wilson v, Table of n, a(n) for n = 1..100
J. P. M. Binet, Mémoire sur les intégrales définites Eulériennes et sur leur application à la théorie des suites ainsi qu'à l`évaluation des functions des grands nombres, Journal de l`École Polytechnique, XVI:123-343, July 1839.
Ch. Hermite, Sur la function log Gamma(a) Journal für die reine und angewandte Mathematik, 115:201-208, 1895.
G. Nemes, Generalization of Binet's Gamma function formulas, Integral Transforms and Special Functions, 24(8):595-606, 2013.
Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv:1602.00336 [math.NT], 2016.
P. Van Mieghem, Binet's factorial series and extensions to Laplace transforms, arXiv:2102.04891 [math.FA], 2021.
Wikipedia, Stirling's Approximation

Cf. A122253 (denominators), A001163, A001164.

r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j=1..n)/(2*n):
a := n -> numer(r(n)); seq(a(n), n=1..21); # Peter Luschny, Sep 22 2021

Rising[z_, n_Integer/;n>0] := z Rising[z + 1, n - 1]; Rising[z_, 0] := 1; c[n_Integer/;n>0] := Integrate[Rising[x, n] (x - 1/2), {x, 0, 1}] / n; Numerator@ Array[c, 19] (* updated by Robert G. Wilson v, Aug 15 2015 *)

a(n) = numerator(sum(j=1, n, (-1)^(n-j)*stirling(n,j,1)*j/((j+1)*(j+2)))/(2*n)); \\ Michel Marcus, Sep 22 2021

a(n) = numerator(c(n)), where c(n) are given by Binet's formulas:
log Gamma z = (z - 1/2) log z - z + log(2*Pi)/2 + Sum_{n >= 1} c(n)/(z+1)^(n), where z^(n) is the rising factorial.
c(n) = (1/n)*Integral_{x=0..1} x^(n)*(x - 1/2).
a(n) = numerator((1/2n)*Sum_{j=1..n} (-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2))). - Peter Luschny, Sep 22 2021

Edited by Peter Luschny, Sep 22 2021

cp's OEIS Frontend

A182913 Denominators of an asymptotic series for the Gamma function (G. Nemes).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

A182935 Numerators of an asymptotic series for the factorial function (Stirling's formula with half-shift).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Formula

A321938 Denominators of the Maclaurin coefficients of exp(1/x - 1/(exp(x)-1) - 1/2).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

A362113 Truncate Stirling's asymptotic series for 1! after n terms and round to the nearest integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Formula

A362116 Truncate Stirling's asymptotic series for 4! after n terms and round to the nearest integer.

Views

Author

Keywords

Comments

Links

Crossrefs

Extensions

A097301 Numerators of rationals used in the Euler-Maclaurin type derivation of Stirling's formula for N!.

Views

Author

Keywords

Comments