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A046968 Numerators of coefficients in Stirling's expansion for log(Gamma(z)). 10
1, -1, 1, -1, 1, -691, 1, -3617, 43867, -174611, 77683, -236364091, 657931, -3392780147, 1723168255201, -7709321041217, 151628697551, -26315271553053477373, 154210205991661, -261082718496449122051, 1520097643918070802691 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,6

COMMENTS

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

REFERENCES

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

LINKS

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.

FORMULA

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

MAPLE

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

MATHEMATICA

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-Fran├žois Alcover, Jun 13 2017 *)

PROG

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

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

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

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

CROSSREFS

Denominators given by A046969.

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

Sequence in context: A141588 A281331 A281332 * A255505 A001067 A141590

Adjacent sequences:  A046965 A046966 A046967 * A046969 A046970 A046971

KEYWORD

frac,sign,nice

AUTHOR

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

EXTENSIONS

More terms from Frank Ellermann, Jun 13 2001

STATUS

approved

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Last modified April 3 19:43 EDT 2020. Contains 333198 sequences. (Running on oeis4.)