A046969 -id:A046969 - cp's OEIS frontend

A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691

Offset: 1

Douglas Stoll (dougstoll(AT)email.msn.com)

A001067(n) = a(n) if n<574; A001067(574) = 37*a(574). - Michael Somos, Feb 01 2004

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. 6.1.41.
L. V. Ahlfors, Complex Analysis, McGraw-Hill, 1979, p. 205

Seiichi Manyama, Table of n, a(n) for n = 1..314
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. 6.1.41.
R. P. Brent, Asymptotic approximation of central binomial coefficients with rigorous error bounds, arXiv:1608.04834 [math.NA], 2016.
N. Elezovic, Asymptotic Expansions of Central Binomial Coefficients and Catalan Numbers, J. Int. Seq. 17 (2014) # 14.2.1.
Eric Weisstein's World of Mathematics, Stirling's Series
Index entries for sequences related to Bernoulli numbers.

Denominators given by A046969.

Similar to but different from A001067. See A090495, A090496.

List([1..25], n-> NumeratorRat(Bernoulli(2*n)/(2*n*(2*n-1))) ); # G. C. Greubel, Sep 19 2019

[Numerator(Bernoulli(2*n)/(2*n*(2*n-1))): n in [1..25]]; // G. C. Greubel, Sep 19 2019

seq(numer(bernoulli(2*n)/(2*n*(2*n-1))), n = 1..25); # G. C. Greubel, Sep 19 2019

Table[ Numerator[ BernoulliB[2n]/(2n(2n - 1))], {n, 1, 22}] (* Robert G. Wilson v, Feb 03 2004 *)
s = LogGamma[z] + z - (z - 1/2) Log[z] - Log[2 Pi]/2 + O[z, Infinity]^42; DeleteCases[CoefficientList[s, 1/z], 0] // Numerator (* Jean-François Alcover, Jun 13 2017 *)

a(n)=if(n<1,0,numerator(bernfrac(2*n)/(2*n)/(2*n-1)))

[numerator(bernoulli(2*n)/(2*n*(2*n-1))) for n in (1..25)] # G. C. Greubel, Sep 19 2019

From numerator of Jk(z) = (-1)^(k-1)*Bk/(((2k)*(2k-1))*z^(2k-1)), so Gamma(z) = sqrt(2*Pi)*z^(z-0.5)*exp(-z)*exp(J(z)).

More terms from Frank Ellermann, Jun 13 2001

A230282 Largest k such that (k*n)! >= (k!)^(n+1).

Original entry on oeis.org

1, 1, 6, 64, 679, 8468, 126784, 2238565, 45605124, 1053117974, 27182818156, 775557529509, 24236473829015, 823299898542083, 30205566231626957, 1190319005015526817, 50143449209799256306, 2248672171655330927835

Offset: 0

Alex Ratushnyak, Oct 14 2013

nonn

			Biggest k such that (3*k)! >= k!^4 is k = 64, so a(3) = 64.
a(10) = 27182818156 because k = 27182818156 satisfies the inequality (k*10)! >= (k!)^11, but k = 27182818157 does not. To verify this, note that taking the logarithm of each side of the inequality gives log((k*10)!) >= 11*log(k!), and use the series expression log(m!) = log(2*Pi*m)/2 + m*log(m) - m + (1/12)/m - (1/360)/m^3 + (1/1260)/m^5 - ... (where the numerators and denominators of the fractions 1/12, -1/360, 1/1260, etc., are from A046968 and A046969, respectively), to get, at k = 27182818156, log(271828181560!) = 6884982704601.26... for the left hand side of the inequality, and the slightly smaller result 11*log(27182818156!) = 6884982704600.83... for the right hand side; then repeat the calculations using k = 27182818157, and observe that this makes the right hand side slightly larger than the left hand side. - _Jon E. Schoenfield_, Oct 23 2013

Cf. A000142, A065027, A136432, A230319.

Table[k = 0; While[(k n)! >= (k!)^(n + 1), k++]; k - 1, {n, 0, 4}] (* T. D. Noe, Oct 18 2013 *)

import math
for n in range(8):
  for k in range(10000000):
    if math.factorial(n*k) < math.factorial(k)**(n+1):
      print(k-1, end=', ')
      break

For n > 1, a(n) = floor(e*(n^n) - ((n^2-1)*log(n) + n*(1+log(2*Pi)))/2) [conjectural, but verified for all n in 2..5000]. - Jon E. Schoenfield, Oct 22 2013

a(7)-a(17) from Jon E. Schoenfield, Oct 22 2013

A127585 Exponential error term from Stirling's Approximation.

Original entry on oeis.org

1, 1, 18, 345, 10243, 437769, 25260317, 1873346813, 172254143084, 19114537903943, 2506628271002200, 382005168783773474, 66734799966312471195, 13212509243902296154744, 2936153006332857671962341, 726345521215072990990045577, 198595552305314906351047196508

Offset: 0

Jonathan Vos Post, Apr 02 2007

			a(1) = Floor[(sqrt(2*pi) * (1^1) * (1^(1/2))) - 1! ] = Floor(1.50662827) = 1.
a(2) = Floor[(sqrt(2*pi) * (2^2) * (2^(2/2))) - 2! ] = Floor(18.0530262) = 18.

Eric Weisstein's World of Mathematics, Stirling's Series.
Eric Weisstein's World of Mathematics, Stirling's Approximation.

Cf. A005394, A046968, A046969, A055775, A127426.

a(n) = floor(sqrt(2*Pi)*(n^n)*(n^(n/2))) - n!.

More terms from Alois P. Heinz, Jan 24 2024

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A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)).

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A230282 Largest k such that (k*n)! >= (k!)^(n+1).

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A127585 Exponential error term from Stirling's Approximation.

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