A001164 -id:A001164 - cp's OEIS frontend

A260447 Numerators in the asymptotic expansion of the Barnes G-function.

1, -1, -1, 157, 65911, -918227, -234932171, 592642957, 149463069056137, -16648792617135289, -286497627060094895989, 3538603031642540133299, 57674522110226646157873673, -713986824035720029666660757, -6478234620955890989297122598683

Offset: 0

Vladimir Reshetnikov, Nov 10 2015

G. C. Greubel and D. Turner, Table of n, a(n) for n = 0..175
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A260448 (denominators), A000178, A001163, A001164.

Numerator[Exp[Series[LogBarnesG[x] - 1/12 - x + 3 x^2/4 + Log[Glaisher] + Log[2 Pi]/2 - x Log[2 Pi]/2 - 5 Log[x]/12 + x Log[x] - x^2 Log[x]/2, {x, Infinity, 20}]][[3]]]

G(x) ~ exp^(-3*x^2/4 + x + zeta'(-1)) * x^(x^2/2 - x + 5/12) * (2*Pi)^((x-1)/2) * (1 + (-1/12)/x + (-1/1440)/x^2 + (157/51840)/x^3 + (65911/87091200)/x^4 + ...).

A260448 Denominators in the asymptotic expansion of the Barnes G-function.

Original entry on oeis.org

1, 12, 1440, 51840, 87091200, 1045094400, 376233984000, 902961561600, 166867296583680000, 18021668031037440000, 140569010642092032000000, 1686828127705104384000000, 8501613763633726095360000000, 102019365163604713144320000000, 208119504933753614814412800000000

Offset: 0

Vladimir Reshetnikov, Nov 10 2015

10^(2m)|a(n) where 5m <= n <= 5m+4, m>=0. Example: for m=4, 20<= n <= 24, the values of a(20) to a(24) are divisible by 10^(10). - G. C. Greubel, Dec 15 2015

G. C. Greubel and D. Turner, Table of n, a(n) for n = 0..175
Eric Weisstein's World of Mathematics, Barnes G-Function.
Wikipedia, Barnes G-function.

Cf. A260447 (numerators), A000178, A001163, A001164.

Denominator[Exp[Series[LogBarnesG[x] - 1/12 - x + 3 x^2/4 + Log[Glaisher] + Log[2 Pi]/2 - x Log[2 Pi]/2 - 5 Log[x]/12 + x Log[x] - x^2 Log[x]/2, {x, Infinity, 20}]][[3]]]

G(x) ~ exp^(-3*x^2/4 + x + zeta'(-1)) * x^(x^2/2 - x + 5/12) * (2*Pi)^((x-1)/2) * (1 + (-1/12)/x + (-1/1440)/x^2 + (157/51840)/x^3 + (65911/87091200)/x^4 + ...).

A272097 Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.

Original entry on oeis.org

1, 0, 0, 2, 6, 8, 7, 9, 1, 3, 2, 4, 1, 5, 2, 7, 9, 4, 1, 5, 8, 4, 3, 4, 5, 5, 4, 6, 4, 3, 4, 5, 2, 0, 9, 6, 1, 8, 1, 8, 1, 0, 4, 0, 3, 1, 9, 2, 3, 6, 7, 8, 8, 8, 3, 7, 2, 8, 6, 6, 5, 6, 7, 3, 8, 0, 6, 4, 7, 7, 8, 5, 0, 6, 2, 1, 1, 1, 0, 0, 7, 3, 8, 5, 3, 8, 1, 0, 9, 5, 8, 8, 6, 6, 7, 8, 2, 6, 3, 5, 8, 8, 0, 1, 9

Offset: 1

Vaclav Kotesovec, Apr 20 2016

Product_{k=1..n} (k! / (sqrt(2*Pi*k) * k^k * exp(-k))) ~ c * n^(1/12), where c = exp(1/12)*(2*Pi)^(1/4) / A^2 = A213080 = 1.04633506677050318098095065697776..., where A = A074962 is the Glaisher-Kinkelin constant.

Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n) + 1/(288*n^2)))) = exp(1/12) * (2*Pi)^(1/4) * abs(Gamma(25/24 + i/24))^2 / A^2 = 0.997305599490607358564533726617761207426462854447669845..., where A = A074962 is the Glaisher-Kinkelin constant and i is the imaginary unit.

			1.00268791324152794158434554643452096181810403192367888372866567380647785...

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A000142, A001163, A001164, A074962, A203138, A203140, A213080, A241140.

Product[n!/(n^n/E^n*Sqrt[2*Pi*n]*(1 + 1/(12*n))), {n, 1, Infinity}]
RealDigits[E^(1/12)*(2*Pi)^(1/4)*Gamma[13/12]/Glaisher^2, 10, 120][[1]]

Product_{n>=1} (n! / (sqrt(2*Pi*n) * n^n * exp(-n) * (1 + 1/(12*n)))).

Equals exp(1/12) * (2*Pi)^(1/4) * Gamma(1/12) / (12 * A^2), where A = A074962 is the Glaisher-Kinkelin constant.

A122253 Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.

Original entry on oeis.org

12, 12, 360, 60, 280, 168, 5040, 180, 11880, 264, 240240, 10920, 13104, 720, 367200, 3060, 813960, 15960, 1053360, 27720, 3825360, 16560, 5023200, 163800, 982800, 3024, 2630880, 6960, 33227040, 229152, 116867520, 235620, 282744, 2520, 1612119600, 7676760, 46060560

Offset: 1

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

See A122252 for references and formulas.

			c(1) = (1/1)*Integral_{x=0..1} x*(x - 1/2) dx = Integral_{x=0..1} (x^2 - x/2) dx = (x^3/3 - x^2/4)|{x=1} - (x^3/3 - x^2/4)|{x=0} = 1/12.

Raphael Schumacher, Rapidly Convergent Summation Formulas involving Stirling Series, arXiv preprint arXiv:1602.00336 [math.NT], 2016.
Wikipedia, Stirling's approximation

Cf. A122252 (numerators), A001163, A001164.

r := n -> add((-1)^(n-j)*Stirling1(n,j)*j/((j+1)*(j+2)), j = 1..n)/(2*n):
a := n -> denom(r(n)); seq(a(n), n = 1..37); # 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;

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

c(n) = (1/n)*Integral_{x=0..1} x^n*(x - 1/2) dx.

A158503 Triangle read by rows: numerators of coefficients of the polynomials phi_s(t) used for asymptotic elementary function expansions of parabolic cylinder functions U(a, x), V(a, x).

Original entry on oeis.org

1, -9, -30, -20, 945, 8028, 19404, 18480, 6160, -1403325, -20545650, -94064328, -200166120, -220540320, -122522400, -27227200, 820945125, 17610977880, 124110533448, 431932849920, 857710030320, 1023307084800, 728175127680, 285558873600, 47593145600

Offset: 0

Chris Kormanyos (ckormanyos(AT)yahoo.com), Mar 20 2009

Each polynomial phi_s(t) has 2s+1 terms. The signs of the polynomials alternate with s with positive coefficients for s even and negative coefficients for s odd.

			The polynomials phi_0, phi_1, phi_2 and phi_3 are:
  1
  -(t/12)*(9 + 30*t + 20*t^2)
  (t^2/288)*(945 + 8028*t + 19404*t^2 + 18480*t^3 + 6160*t^4)
  -(t^3/51840)*(1403325 + 20545650*t + 94064328*t^2 + 200166120*t^3 + 220540320*t^4 + 122522400*t^5 + 27227200*t^6)

Amparo Gil, Javier Segura and Nico M. Temme, Computing the real parabolic cylinder functions U(a, x), V(a, x), ACM TOMS, Volume 32, Issue 1 (March 2006), pages 70-101.
Amparo Gil, Javier Segura and Nico M. Temme, Numerical Methods for Special Functions, SIAM, 2007, pages 378-385. See Equations 12.121 through 12.125

Chris Kormanyos, Rows s = 0..122, flattened

For denominators see A001164.

pktop = {1, -9, -30, -20};
pkbot = {1, 12};
p = (-t/12) (9 + (30 t) + (20 (t^2)));
Do[pk = -(4 (t^2) ((t + 1)^2)) D[p, t] - ((1/4) Integrate[((20 (t^2)) + (20 t) + 3) p, {t, 0, t}]);
p = Together[Simplify[pk]];
Do[pktop = Append[pktop, Coefficient[Expand[Numerator[p]], t^n]], {n, k, (2 k) + k, 1}];
pkbot = Append[pkbot, Denominator[p]];
Print[k], {k, 2, 10, 1}];

phi_s+1(t) = ( -4*t^2*(t + 1)^2 * d/dt[phi_s(t)] ) - (1/4)*Integral_{T=0..t} (20*T^2 + 20*T + 3)*phi_s(T) dT

phi_0 = 1, phi_-1 = 0

A177677 The maximum integer dimension in which the volume of the hypersphere of radius n remains larger than 1.

Original entry on oeis.org

12, 62, 147, 266, 419, 607, 828, 1084, 1375, 1699, 2057, 2450, 2877, 3338, 3833, 4362, 4926, 5523, 6155, 6821, 7521, 8256, 9024, 9827, 10664, 11535, 12440, 13379, 14353, 15360, 16402, 17478, 18588, 19732, 20911, 22123, 23370, 24651, 25966, 27315

Offset: 1

Michel Lagneau, May 10 2010

nonn

The volume of the d-dimensional hypersphere of radius n is V= Pi^(d/2) * n^d / Gamma(1 + d/2).

For fixed radius, V -> 0 as d->infinity, so there is a dimension d for which V(n,d) > 1 but V(n,d+1) < 1, which defines the entry in the sequence.

			a(n=2)=62 because Pi^(62/2) * 2^62/GAMMA(1 + (62/2)) =1.447051 and Pi^(63/2)* 2^63 / Gamma(1 + (63/2)) =0.9103541.

Eric Weisstein, Stirling Series, MathWorld.
Wikipedia, Hypersphere

Cf. A005446 , A005147, A001164, A005146, A005447, A001163

with(numtheory): n0:=50: T:=array(1..n0): for r from 1 to n0 do: x:=2: for n from 1 to 1000000 while(x>=1) do: x:= floor(evalf((r^n * Pi^(n/2))/GAMMA(1 + n/2))):k:=n:od:T[r]:=k-1:od:print(T):

a(n) = max {d: Pi^d/2 * n^d / Gamma(1+d/2) > 1}.

Use of variables standardized. Definition simplified, comments tightened, unspecific reference and superfluous parentheses removed - R. J. Mathar, Oct 20 2010

A362114 Truncate Stirling's asymptotic series for 2! after n terms and round to the nearest integer.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 1, 6, 6, -37, -38, 388, 406, -4245, -4439, 51286, 53609, -676862, -707319, 9728466, 10163563, -151746398, -158496540, 2560628713, 2673985990, -46607156242, -48661551041, 912537268911

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-A362116.

A362115 Truncate Stirling's asymptotic series for 3! after n terms and round to the nearest integer.

Original entry on oeis.org

6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 9, 9, -21, -22, 246, 253, -2243, -2313, 22563, 23254, -241529, -248887, 2755875, 2839370, -33440529, -34448635, 430760619

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-A362116.

A154268 Ratios of consecutive denominators of Stirling's expansion for the Gamma function.

Original entry on oeis.org

12, 24, 180, 48, 84, 360, 12, 96, 5940, 168, 156, 720, 12, 24, 107100, 192, 228, 11880, 12, 336, 4140, 312, 12, 1440, 84, 24, 516780, 48, 372, 214200, 12

Offset: 0

Paul Curtz, Jan 06 2009

The sequence is finite because the ratios are no longer integer after a(30).
All entries are multiples of 12.
a(n) mod 9 is one of {0,3,6}.

dd = Denominator[ CoefficientList[ Normal[ Series[ E^x*x^(x-1/2)*x! / Sqrt[2*Pi], {x, Infinity, 31}]] /. x -> 1/x, x]]; Rest[ dd / RotateRight[dd] ] (* Jean-François Alcover, Aug 03 2012 *)

a(n) = A001164(n+1)/A001164(n).

Edited and extended by R. J. Mathar, Sep 07 2009

cp's OEIS Frontend

A260447 Numerators in the asymptotic expansion of the Barnes G-function.

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A260448 Denominators in the asymptotic expansion of the Barnes G-function.

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A272097 Decimal expansion of an infinite product involving the ratio of n! to its Stirling approximation.

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A122253 Binet's factorial series. Denominators of the coefficients of a convergent series for the logarithm of the Gamma function.

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A158503 Triangle read by rows: numerators of coefficients of the polynomials phi_s(t) used for asymptotic elementary function expansions of parabolic cylinder functions U(a, x), V(a, x).

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A177677 The maximum integer dimension in which the volume of the hypersphere of radius n remains larger than 1.

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Maple

Formula

Extensions

A362114 Truncate Stirling's asymptotic series for 2! after n terms and round to the nearest integer.

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A362115 Truncate Stirling's asymptotic series for 3! after n terms and round to the nearest integer.

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