A262235 -id:A262235 - cp's OEIS frontend

A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.

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

Offset: 0

N. J. A. Sloane

Yee (2010) computed 29844489545 decimal digits of gamma.
Decimal expansion of 0th Stieltjes constant. - Paul Muljadi, Aug 24 2010
The value of Euler's constant is close to (18/Pi^2)*Sum_{n>=0} 1/4^(2^n) = 0.5770836328... = (6/5) * A082020 * A078585. - Arkadiusz Wesolowski, Mar 27 2012

			0.577215664901532860606512090082402431042...

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 3.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 24, 259-262.
S. R. Finch, Mathematical Constants, Encyclopedia of Mathematics and its Applications, vol. 94, Cambridge University Press, 2003, pp. 28-40, 166, 365.
C. F. Gauss, Disquisitiones Arithmeticae, Yale, 1965; see p. 359.
B. Gugger, Problèmes corrigés de Mathématiques posés aux concours des Ecoles Militaires, Ecole de l'Air, 1992, option MP, 1ère épreuve, Ellipses, 1993, pp. 167-184.
Jan Gullberg, Mathematics from the Birth of Numbers, W. W. Norton & Co., NY & London, 1997, §8.3 Infinite Series, pp. 273-274.
J. Havil, Gamma: Exploring Euler's Constant, Princeton Univ. Press, 2003.
J.-M. Monier, Analyse, Exercices corrigés, 2ème année, MP, Dunod, Exercice 4.3.14, pages 371 and 387, 1997.
Paulo Ribenboim, The Little Book of Bigger Primes, Springer-Verlag NY 2004. See p. 166.
Joel L. Schiff, The Laplace Transform: Theory and Applications, Springer-Verlag New York, Inc. (1999). See p. 44.
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 1, equation 1:7:5 at page 13.
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987, p. 28.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, Cambridge Univ. Press, 4th ed., 1990.

Harry J. Smith, Table of n, a(n) for n = 0..20000
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].
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/Pi^2 and into the formal enveloping series with rational coefficients only, arXiv:1501.00740 [math.NT], 2015-2016; Journal of Number Theory (Elsevier), Volume 158, pages 365-396, 2016.
D. Bradley, Ramanujan's formula for the logarithmic derivative of the Gamma function, arXiv:math/0505125 [math.CA], 2005.
R. P. Brent, Ramanujan and Euler's constant.
R. P. Brent and F. Johansson, A bound for the error term in the Brent-McMillan algorithm, arXiv 1312.0039 [math.NA], Nov. 2013.
C. K. Caldwell, The Prime Glossary, Euler's constant.
D. Castellanos, The ubiquitous pi, Math. Mag., 61 (1988), 67-98 and 148-163.
Chao-Ping Chen, Sharp inequalities and asymptotic series of a product related to the Euler-Mascheroni constant, Journal of Number Theory, Volume 165, August 2016, Pages 314-323.
E. Chlebus, A recursive scheme for improving the original rate of convergence to the Euler-Mascheroni constant, Amer. Math. Mnthly, 118 (2011), 268-274.
M. Coffey and Jonathan Sondow, Rebuttal of Kowalenko's paper as concerns the irrationality of Euler's constant, arXiv:1202.3093 [math.NT], 2012; Acta Appl. Math., 121 (2012), 1-3.
Dave's Math Tables, Gamma Constant.
Philippe Deléham, Letter to N. J. A. Sloane, Apr 14 1997.
Thomas and Joseph Dence, A survey of Euler's constant, Math. Mag., 82 (2009), 255-265.
Pierre Dusart, Explicit estimates of some functions over primes, The Ramanujan Journal, 2016.
Ph. Flajolet and I. Vardi, Zeta function expansions of some classical constants.
X. Gourdon and P. Sebah, The Euler constant gamma.
Kalpok Guha and Sourangshu Ghosh, Measuring Abundance with Abundancy Index, (2021).
Brady Haran and Tony Padilla, The mystery of 0.577, Numberphile video, 2016.
J. C. Kluyver, Euler's constant and natural numbers, Proc. K. Ned. Akad. Wet., 27(1-2) (1924), 142-144.
D. E. Knuth, Euler's constant to 1271 places, Math. Comp. 16 1962 275-281.
Stefan Krämer, Euler's Constant γ=0.577... Its Mathematics and History.
Richard Kreckel, 116 million digits of Euler's constant (bzipped).
A. Krowne, PlanetMath.org, Euler's constant.
Jeffrey C. Lagarias, Euler's constant: Euler's work and modern developments, arXiv:1303.1856 [math.NT], 2013; Bull. Amer. Math. Soc., 50 (2013), 527-628.
M. Lerch, Expressions nouvelles de la constante d'Euler, S.-B. Kgl. Bohmischen Ges. Wiss., Article XLII (1897), Prague (5 pages).
Eric Naslund, Euler-Mascheroni constant expression, further simplification, MathStackExchange.
T. Papanikolaou, Plouffe's Inverter, Euler's constant to 1000000 decimals.
Michael Penn, Euler's other constant, YouTube video (2023).
S. Plouffe, using data from J. Borwein, 170000 digits of Euler or gamma constant [archived copy of a page on WorldWideSchool.org which doesn't exist any mode, cf. "Euler" link in left column].
S. Ramanujan, A series for Euler's constant, Messenger of Math., 46 (1917), 73-80.
S. Ramanujan, Question 327, J. Ind. Math. Soc.
Jonathan Sondow, An antisymmetric formula for Euler's constant, Math. Mag. 71 (1998), 219-220.
Jonathan Sondow, Criteria for irrationality of Euler's constant, Proc. Amer. Math. Soc. 131 (2003), 3335-3344.
Jonathan Sondow, Double integrals for Euler's constant and ln(4/Pi) and an analog of Hadjicostas's formula, arXiv:math/0211148 [math.CA], 2002-2004; Amer. Math. Monthly 112 (2005), 61-65.
Jonathan Sondow, An infinite product for e^gamma via hypergeometric formulas for Euler's constant, gamma, arXiv:math/0306008 [math.CA], 2003.
Jonathan Sondow, A hypergeometric approach, via linear forms involving logarithms, to irrationality criteria for Euler's constant. With an Appendix by Sergey Zlobin, arXiv:math/0211075 [math.NT], 2002-2009; Math. Slovaca 59 (2009), 1-8.
Jonathan Sondow, New Vacca-Type Rational Series for Euler's Constant and Its "Alternating" Analog ln(4/Pi), arXiv:math/0508042 [math.NT], 2005; Additive Number Theory, Festschrift In Honor of the Sixtieth Birthday of Melvyn B. Nathanson (D. Chudnovsky and G. Chudnovsky, eds.), Springer, 2010, pp. 331-340.
Jonathan Sondow and Petros Hadjicostas, The generalized-Euler-constant function gamma(z) and a generalization of Somos's quadratic recurrence constant, arXiv:math/0610499 [math.CA], 2006; J. Math. Anal. Appl. 332 (1) (2007), 292-314.
Jonathan Sondow and S. Zlobin, Integrals over polytopes, multiple zeta values and polylogarithms, and Euler's constant, arXiv:0705.0732 [math.NT], 2007; Math. Notes, 84 (2008), 568-583, Erratum p. 887.
Jonathan Sondow and W. Zudilin, Euler's constant, q-logarithms and formulas of Ramanujan and Gosper, arXiv:math/0304021 [math.NT], 2003; Ramanujan J. 12 (2006), 225-244.
D. W. Sweeney, On the computation of Euler's constant, Math. Comp., 17 (1963), 170-178.
Eric Weisstein's World of Mathematics, Euler-Mascheroni Constant.
Wikipedia, Stieltjes constants.
A. Y. Yee, Large computations.
Index entries for sequences related to Beatty sequences.

Cf. A002852 (continued fraction).
Cf. A073004 (exp(gamma)) and A094640 ("alternating Euler constant").
Cf. A231095 (power tower using this constant).
Cf. A199332, A252898.
Denote the generalized Euler constants, also called Stieltjes constants, by Sti(n).
Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
Sti(1/2) = A301816, Sti(1) = A082633 (cf. A262382/A262383), Sti(3/2) = A301817,
Sti(2) = A086279 (cf. A262384/A262385), Sti(3) = A086280 (cf. A262386/A262387),
Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,
Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

EulerGamma(250); // G. C. Greubel, Aug 21 2018

Maple
```
Digits := 100; evalf(gamma);
```

RealDigits[ EulerGamma, 10, 105][[1]] (* Robert G. Wilson v, Nov 01 2004 *)
(1/2) N[Sum[PolyGamma[0, 1/2 + 2^k] - PolyGamma[0, 2^k], {k, 0, Infinity }], 30] (* Dimitri Papadopoulos, Nov 30 2016 *)

default(realprecision, 20080); x=Euler; d=0; for (n=0, 20000, x=(x-d)*10; d=floor(x); write("b001620.txt", n, " ", d));  \\ Harry J. Smith, Apr 15 2009

from sympy import S
def aupton(digs): return [int(d) for d in str(S.EulerGamma.n(digs+2))[2:-2]]
print(aupton(99)) # Michael S. Branicky, Nov 22 2021

Limit_{n->oo} (1 + 1/2 + ... + 1/n - log(n)) (definition).
Sum_{n>=1} (1/n - log(1 + 1/n)), since log(1 + 1/1) + ... + log(1 + 1/n) telescopes to log(n+1) and lim_{n->infinity} (log(n+1) - log(n)) = 0.
Integral_{x=0..1} -log(log(1/x)). - Robert G. Wilson v, Jan 04 2006
Integral_{x=0..1,y=0..1} (x-1)/((1-x*y)*log(x*y)). - (see Sondow 2005)
Integral_{x=0..oo} -log(x)*exp(-x). - Jean-François Alcover, Mar 22 2013
Integral_{x=0..1} (1 - exp(-x) - exp(-1/x))/x. - Jean-François Alcover, Apr 11 2013
Equals the lim_{n->oo} fractional part of zeta(1+1/n). The corresponding fractional part for x->1 from below, using n-1/n, is -(1-a(n)). The fractional part found in this way for the first derivative of Zeta as x->1 is A252898. - Richard R. Forberg, Dec 24 2014
Limit_{x->1} (Zeta(x)-1/(x-1)) from Whittaker and Watson. 1990. - Richard R. Forberg, Dec 30 2014
exp(gamma) = lim_{i->oo} exp(H(i)) - exp(H(i-1)), where H(i) = i-th Harmonic number. For a given n this converges faster than the standard definition, and two above, after taking the logarithm (e.g., 13 digits vs. 6 digits at n=3000000 or x=1+1/3000000). - Richard R. Forberg, Jan 08 2015
Limit_{n->oo} (1/2) Sum_{j>=1} Sum_{k=1..n} ((1 - 2*k + 2*n)/((-1 + k + j*n) (k + j*n))). - Dimitri Papadopoulos, Jan 13 2016
Equals 25/27 minus lim_{x->oo} 2^(x+1)/3 - (22/27)*(4/3)^x - Zeta(Sum_{i>=1} (H_i/i^x)), letting H_i denote the i-th harmonic number. - John M. Campbell, Jan 29 2016
Limit_{x->0} -B'(x), where B(x) = -x zeta(1-x) is the "Bernoulli function". - Jean-François Alcover, May 20 2016
Sum_{k>=0} (1/2)(digamma(1/2+2^k) - digamma(2^k)) where digamma(x) = d/dx log(Gamma(x)). - Dimitri Papadopoulos, Nov 14 2016
Using the abbreviations a = log(z^2 + 1/4)/2, b = arctan(2*z) and c = cosh(Pi*z) then gamma = -Pi*Integral_{0..oo} a/c^2. The general case is for n >= 0 (which includes Euler's gamma as gamma_0) gamma_n = -(Pi/(n+1))* Integral_{0..oo} sigma(n+1)/c^2, where sigma(n) = Sum_{k=0..floor(n/2)} (-1)^k*binomial(n,2*k)*b^(2*k) *a^(n-2*k). - Peter Luschny, Apr 19 2018
Limit_{s->0} (Zeta'(1-s)*s - Zeta(1-s)) / (Zeta(1-s)*s). - Peter Luschny, Jun 18 2018
log(2) * (gamma - (1/2) * log(2)) = -Sum_{v >= 1} (1/2^(v+1)) * (Delta^v (log(w)/w))|{w=1}, where Delta(f(w)) = f(w) - f(w + 1) (forward difference). [This is a formula from Lerch (1897).] - _Petros Hadjicostas, Jul 21 2019
From Amiram Eldar, Jul 05 2020: (Start)
Equals Integral_{x=1..oo} (1/floor(x) - 1/x) dx.
Equals Integral_{x=0..1} (1/(1-x) + 1/log(x)) dx = Integral_{x=0..1} (1/x + 1/log(1-x)) dx.
Equals -Integral_{-oo..oo} x*exp(x-exp(x)) dx.
Equals Sum_{k>=1} (-1)^k * floor(log_2(k))/k.
Equals (-1/2) * Sum_{k>=1} (Lambda(k)-1)/k, where Lambda is the Mangoldt function. (End)
Equals Integral_{0..1} -1/LambertW(-1,-x*exp(-x)) dx = 1 + Integral_{0..1} LambertW(-1/x*exp(-1/x)) dx. - Gleb Koloskov, Jun 12 2021
Equals Sum_{k>=2} (-1)^k * zeta(k)/k. - Vaclav Kotesovec, Jun 19 2021
Equals lim_{x->oo} log(x) - Sum_{p prime <= x} log(p)/(p-1). - Amiram Eldar, Jun 29 2021
Limit_{n->oo} (2*HarmonicNumber(n) - HarmonicNumber(n^2)). After answer by Eric Naslund on Mathematics Stack Exchange, on Jun 21 2011. - Mats Granvik, Jul 19 2021
Equals Integral_{x=0..oo} ( exp(-x) * (1/(1-exp(-x)) - 1/x) ) dx (see Gugger or Monier). - Bernard Schott, Nov 21 2021
Equals 1/2 + Limit_{s->1} (Zeta(s) + Zeta(1/s))/2. - Thomas Ordowski, Jan 12 2023
Equals Sum_{j>=2} Sum_{k>=2} ((k-1)/(k*j^k)). - Mike Tryczak, Apr 06 2023
From Stefano Spezia, Oct 27 2024: (Start)
Equals Sum_{n>=1} n*(zeta(n+1) - 1)/(n + 1) [Euler] (see Finch at p. 30).
Equals lim_{n->oo} Sum_{prime p<=n} log(p/(p - 1)) - log(log(n)) (see Finch at p. 31). (End)
Equals lim_{s->1} zeta(s) - zeta(s)^2/zeta(2*s - 1)/2. - Mats Granvik, Jul 07 2025

A002657 Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

Original entry on oeis.org

1, 1, 5, 9, 251, 475, 19087, 36799, 1070017, 2082753, 134211265, 262747265, 703604254357, 1382741929621, 8164168737599, 5362709743125, 8092989203533249, 15980174332775873, 12600467236042756559

Offset: 0

N. J. A. Sloane

These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers. [After the Danish mathematician Niels Erik Nørlund (1885-1981). - Amiram Eldar, Jun 17 2021]
The denominators are found in A002790. The alternating rational sequence ((-1)^n)*a(n)/A002790(n)is the z-sequence for the Stirling2 triangle A008277(n+1,k+1), n>=k>=0. This is the Sheffer (exp(x),exp(x)-1) triangle. See the W. Lang link under A006232 for Sheffer a- and z-sequences with references, and the conversion to S. Roman's notation. The a-sequence is A006232(n)/A006233(n). - Wolfdieter Lang, Oct 06 2011 [This is the Sheffer triangle A007318*A048993. Added Jun 20 2017]
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015

			1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

Louis Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales, Centre de Calcul Scientifique de l'Armement, Arcueil, 1969.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
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).

T. D. Noe, Table of n, a(n) for n = 0..100
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, 2018, pp. 133-149.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
Donghyun Kim and Jaeseong Oh, Extending the science fiction and the Loehr--Warrington formula, arXiv:2409.01041 [math.CO], 2024. See p. 32.
Takao Komatsu, Convolution Identities for Cauchy Numbers of the Second Kind, Kyushu Journal of Mathematics, Vol. 69, No. 1 (2015), pp. 125-144.
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., Vol. 45, No. 2 (2007), pp. 133-137.
Guo-Dong Liu, H. M. Srivastava, and Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq., Vol. 17 (2014), Article 14.4.6.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math., Vol. 306, No. 16 (2006), pp. 1906-1920.
Louis Melville Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer, 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Michael O. Rubinstein, Identities for the Riemann zeta function, Ramanujan J., Vol. 27, No. 1 (2012), pp. 29-42; arXiv preprint, arXiv:0812.2592 [math.NT], 2008-2009.
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., Vol. 309, No. 12 (2009), pp. 3830-3842.
Index entries for sequences related to Bernoulli numbers.

Cf. A002206, A002207, A002208, A002209, A002790, A006232, A006233, A075266, A075267, A262235.

m:=25; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Numerator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 29 2018

seq(numer(add((-1)^(n-k)*Stirling1(n,k)/(k+1),k=0..n)),n=0..10); # Peter Luschny, Apr 28 2009

Table[Abs[Numerator[NorlundB[n,n]]],{n,0,30}](* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator @ NorlundB[ n, n]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator@Integrate[ Pochhammer[ x, n], {x, 0, 1}]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, Numerator[ n! SeriesCoefficient[ -x / ((1 - x) Log[ 1 - x]), {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)
a[ n_] := If[ n < 0, 0, (-1)^n Numerator[ n! SeriesCoefficient[ (x / (Exp[x] - 1))^n, {x, 0, n}]]]; (* Michael Somos, Jul 12 2014 *)

v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
makelist(num(n!*v(n)),n,0,10); /* Vladimir Kruchinin, Aug 28 2013 */

Numerator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). (Note: the numerator of the coefficient of x^n/n! is a(n). - Michael Somos, Jul 12 2014). E.g.f. rewritten by Iaroslav V. Blagouchine, May 07 2016
Numerator of Sum_{k=0..n} (-1)^(n-k) A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = numerator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013

A002790 Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

Original entry on oeis.org

1, 2, 6, 4, 30, 12, 84, 24, 90, 20, 132, 24, 5460, 840, 360, 16, 1530, 180, 7980, 840, 13860, 440, 1656, 720, 81900, 6552, 216, 112, 3480, 240, 114576, 7392, 117810, 2380, 1260, 72, 3838380, 207480, 32760, 560, 568260, 27720, 238392, 55440, 869400, 2576, 236880

Offset: 0

N. J. A. Sloane

The numerators are given in A002657.
These coefficients (with alternating signs) are also known as the Nørlund [or Norlund, Noerlund or Nörlund] numbers.
A simple series with the signless Cauchy numbers of second type C2(n) leads to Euler's constant: gamma = 1 - Sum_{n >=1} C2(n)/(n*(n+1)!) = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A075266 and A262235. - Iaroslav V. Blagouchine, Sep 15 2015
a(n) appears to be divisible by n+1. - Hal M. Switkay, Aug 15 2025

			1, 1/2, 5/6, 9/4, 251/30, 475/12, 19087/84, 36799/24, 1070017/90, ...

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 294.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951, p. 136.
N. E. Nørlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924.
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).

T. D. Noe, Table of n, a(n) for n = 0..1000
Ibrahim M. Alabdulmohsin, The Language of Finite Differences, in Summability Calculus: A Comprehensive Theory of Fractional Finite Sums, Springer, Cham, pp 133-149.
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016.
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.
C. H. Karlson & N. J. A. Sloane, Correspondence, 1974
Guodong Liu, Some computational formulas for Norlund numbers, Fib. Quart., 45 (2007), 133-137.
Guo-Dong Liu, H. M. Srivastava, Hai-Quing Wang, Some Formulas for a Family of Numbers Analogous to the Higher-Order Bernoulli Numbers, J. Int. Seq. 17 (2014) # 14.4.6.
Rui-Li Liu and Feng-Zhen Zhao, Log-concavity of two sequences related to Cauchy numbers of two kinds, Online Journal of Analytic Combinatorics, Issue 14 (2019), #09.
Donatella Merlini, Renzo Sprugnoli and M. Cecilia Verri, The Cauchy numbers, Discrete Math. 306 (2006), no. 16, 1906-1920.
L. M. Milne-Thompson, Calculus of Finite Differences, 1951. [Annotated scan of pages 135, 136 only]
N. E. Nørlund, Vorlesungen ueber Differenzenrechnung Springer 1924, p. 461.
N. E. Nörlund, Vorlesungen über Differenzenrechnung, Springer-Verlag, Berlin, 1924; page 461 [Annotated scanned copy of pages 144-151 and 456-463]
Feng-Zhen Zhao, Sums of products of Cauchy numbers, Discrete Math., 309 (2009), 3830-3842.
Index entries for sequences related to Bernoulli numbers.

Cf. A002657, A075266, A075267, A262235.

See also A002208, A002209, A002206, A002207, A006232, A006233.

m:=60; R:=PowerSeriesRing(Rationals(), m); b:=Coefficients(R!(-x/((1-x)*Log(1-x)) )); [Denominator(Factorial(n-1)*b[n]): n in [1..m-1]]; // G. C. Greubel, Oct 28 2018

A002790 := proc(n)
    denom(add((-1)^k*Stirling1(n, k)/(k+1), k=0..n)) ;
end proc: # Peter Luschny, Apr 28 2009
v := proc(n) option remember; ifelse(n=0, 1, 1 - add(v(i)/(n-i+1), i=0..n-1)) end:
seq(denom(n!*v(n)), n = 0..46); # after Vladimir Kruchinin, Peter Luschny, Aug 17 2025

Table[ Denominator[ NorlundB[n, n]], {n, 0, 60}] (* Vladimir Joseph Stephan Orlovsky, Dec 30 2010 *)

v(n):=if n=0 then 1 else 1-sum(v(i)/(n-i+1),i,0,n-1);
makelist(denom(n!*v(n)),n,0,10); /* Vladimir Kruchinin, Aug 28 2013 */

Denominator of integral of x(x+1)...(x+n-1) from 0 to 1.
E.g.f.: -x/((1-x)*log(1-x)). - Corrected by Iaroslav V. Blagouchine, May 07 2016.
Denominator of Sum_{k=0..n} (-1)^k A008275(n,k)/(k+1). - Peter Luschny, Apr 28 2009
a(n) = denominator(n!*v(n)), where v(n) = 1 - Sum_{i=0..n-1} v(i)/(n-i+1), v(0)=1. - Vladimir Kruchinin, Aug 28 2013

A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).

Original entry on oeis.org

0, 1, 5, 1, 251, 19, 19087, 751, 1070017, 2857, 26842253, 434293, 703604254357, 8181904909, 1166309819657, 5044289, 8092989203533249, 5026792806787, 12600467236042756559, 69028763155644023, 8136836498467582599787

Offset: 0

Paul D. Hanna, Sep 15 2002

A series with these numerators leads to Euler's constant: gamma = 1 - 1/4 - 5/72 - 1/32 - 251/14400 - 19/1728 - 19087/2540160 - ..., see references [Blagouchine] below, as well as A262235. - Iaroslav V. Blagouchine, Sep 15 2015

Robert Israel, Table of n, a(n) for n = 1..447
Iaroslav V. Blagouchine, Two series expansions for the logarithm of the gamma function involving Stirling numbers and containing only rational coefficients for certain arguments related to 1/pi, Journal of Mathematical Analysis and Applications (Elsevier), 2016. arXiv version, arXiv:1408.3902 [math.NT], 2014-2016
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A053657, A075264, A075267 (denominator), A262235.

S:= series(log(-log(1-x)/x),x,51):
seq(numer(coeff(S,x,j)),j=0..50); # Robert Israel, May 17 2016
# Alternative:
a := proc(n) local r; r := proc(n) option remember; if n=0 then 1 else
1 - add(r(k)/(n-k+1), k=0..n-1) fi end: numer(r(n)/(n*(n+1))) end:
seq(a(n), n=0..20); # Peter Luschny, Apr 19 2018

Numerator[ CoefficientList[ Series[ Log[ -Log[1 - x]/x], {x, 0, 20}], x]]

@cached_function
def r(n): return 1 - sum(r(k)/(n-k+1) for k in range(n)) if n > 0 else 1
def a(n: int): return numerator(r(n)/(n*(n+1))) if n > 0 else 0
print([a(n) for n in range(21)])  # Peter Luschny, Aug 15 2025

a(n) = numerator(Sum_{k=1..n} (k-1)!*(-1)^(n-k-1)*binomial(n,k)*Stirling1(n+k,k)/(n+k)!). - Vladimir Kruchinin, Aug 14 2025

Edited by Robert G. Wilson v, Sep 17 2002

a(0) = 0 prepended by Peter Luschny, Aug 15 2025

A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

Original entry on oeis.org

-1, 11, -137, 121, -7129, 57844301, -1145993, 4325053069, -1848652896341, 48069674759189, -1464950131199, 105020512675255609, -22404210159235777, 1060366791013567384441, -15899753637685210768473787, 2241672100026760127622163469, -8138835628210212414423299

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

			Numerators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

G. C. Greubel, Table of n, a(n) for n = 1..259
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262383 (denominators of this series), A086279, A086280, A262387.

a := n -> numer(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..16); # Peter Luschny, Apr 19 2018

a[n_] := Numerator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = numerator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = numerator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = numerator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

A262383 Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

Original entry on oeis.org

12, 720, 15120, 11200, 332640, 908107200, 4324320, 2940537600, 175991175360, 512143632000, 1427794368, 7795757249280, 107084577600, 279490747536000, 200143324310529600, 1178332991611776000, 157531148611200, 906996615309386784000, 5828652498614400, 262872227687509440000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_1 = - 1/12 + 11/720 - 137/15120 + 121/11200 - 7129/332640 + 57844301/908107200 - ..., see formulas (46)-(47) in the reference below.

			Denominators of -1/12, 11/720, -137/15120, 121/11200, -7129/332640, 57844301/908107200, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382 (numerators of this series), A086279, A086280, A262387.

a := n -> denom(Zeta(1 - 2*n)*(Psi(2*n) + gamma)):
seq(a(n), n=1..20); # Peter Luschny, Apr 19 2018

a[n_] := Denominator[-BernoulliB[2*n]*HarmonicNumber[2*n - 1]/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(-bernfrac(2*n)*sum(k=1,2*n-1,1/k)/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = denominator(-B_{2n}*H_{2n-1}/(2n)), where B_n and H_n are Bernoulli and harmonic numbers respectively.

a(n) = denominator(Zeta(1 - 2*n)*(Psi(2*n) + gamma)), where gamma is Euler's gamma. - Peter Luschny, Apr 19 2018

A262387 Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.

Original entry on oeis.org

1, 120, 1008, 28800, 49896, 101088000, 5702400, 12350257920000, 43480172736000, 7075668600000, 206069667148800, 5919216795588096000, 581222138112000, 8460252005694128640000, 18991807088644406016000, 1150594272774401495040000, 33940540399314092544000, 9737059611553100811150566400000, 1290633707289706940160000, 1263402804161736165764268432000000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.

			Denominators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, 2015.

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384, A262385, A086280, A262386 (numerators of this series).

a[n_] := Denominator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(-bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^3 -3*sum(k=1,2*n-1,1/k)*sum(k=1,2*n-1,1/k^2) + 2*sum(k=1,2*n-1,1/k^3))/(2*n));

a(n) = denominator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2){2n-1}+2*H^(3){2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

A301816 Decimal expansion of the real Stieltjes gamma function at x = 1/2.

Original entry on oeis.org

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

Offset: 0

Peter Luschny, Apr 09 2018

Define the real Stieltjes gamma function (this is not a standard notion) as Sti(x) = -2*Pi*I(x+1)/(x+1) where I(x) = Integral_{-infinity..+infinity} log(1/2+i*z)^x/(exp(-Pi*z) + exp(Pi*z))^2 dz and i is the imaginary unit. We look here at the real part of Sti(x).

			0.2754347245639200799552878777978068357987023238863074873733211475133063441...

Iaroslav V. Blagouchine, A theorem for the closed-form evaluation of the first generalized Stieltjes constant at rational arguments and some related summations, Journal of Number Theory, vol. 148, pp. 537-592 and vol. 151, pp. 276-277, 2015. arXiv version, arXiv:1401.3724 [math.NT], 2014.
Peter Luschny, Illustration of the real Stieltjes gamma function.

Sti(0) = A001620 (Euler's constant gamma) (cf. A262235/A075266),
Sti(1/2) = A301816,
Sti(1) = A082633 (Stieltjes constant gamma_1) (cf. A262382/A262383),
Sti(3/2) = A301817,
Sti(2) = A086279 (Stieltjes constant gamma_2) (cf. A262384/A262385),
Sti(3) = A086280 (Stieltjes constant gamma_3) (cf. A262386/A262387),
Sti(4) = A086281, Sti(5) = A086282, Sti(6) = A183141, Sti(7) = A183167,
Sti(8) = A183206, Sti(9) = A184853, Sti(10) = A184854.

Sti := x -> (-4*Pi/(x + 1))*int(log(1/2 + I*z)^(x + 1)/(exp(-Pi*z) + exp(Pi*z))^2, z=0..64): Sti(1/2): Re(evalf(%, 100)); # Note that this is an approximation which needs a larger domain of integration and higher precision if used for more values than are in the Data section.

c = -Re((4/3)*Pi*Integral_{-oo..oo} log(1/2+i*z)^(3/2)/(exp(-Pi*z)+exp(Pi*z))^2 dz).

A262384 Numerators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

Original entry on oeis.org

0, -1, 5, -469, 6515, -131672123, 63427, -47800416479, 15112153995391, -29632323552377537, 4843119962464267, -1882558877249847563479, 2432942522372150087, -2768809380553055597986831, 334463513629004852735064113, -1125061940756859461946444233539, 333807583501528759350875247323

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.

			Numerators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...

G. C. Greubel, Table of n, a(n) for n = 1..237
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only, Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

The sequence of denominators is A262385.

Cf. A001067, A001620, A002206, A006953, A075266, A082633, A086279, A086280, A195189, A262235, A262382, A262383, A262386, A262387.

a := n -> numer(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..17); # Peter Luschny, Apr 19 2018

a[n_] := Numerator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = numerator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = numerator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

a(n) = numerator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018

A262385 Denominators of a semi-convergent series leading to the second Stieltjes constant gamma_2.

Original entry on oeis.org

1, 60, 336, 21600, 133056, 825552000, 89100, 11435424000, 483113030400, 101889627840000, 1471926193920, 42280119968486400, 3425059028160, 209827678712652000, 1184296360402995840, 163066081742403840000, 1749151741873536000, 20373357051590182072392960000

Offset: 1

Iaroslav V. Blagouchine, Sep 20 2015

gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.

			Denominators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...

G. C. Greubel, Table of n, a(n) for n = 1..500
Iaroslav V. Blagouchine, Expansions of generalized Euler's constants into the series of polynomials in 1/pi^2 and into the formal enveloping series with rational coefficients only. Journal of Number Theory (Elsevier), vol. 158, pp. 365-396, 2016. arXiv version, arXiv:1501.00740 [math.NT], 2015.

Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384 (numerators of this series), A086280, A262386, A262387.

a := n -> denom(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)):
seq(a(n), n=1..18); # Peter Luschny, Apr 19 2018

a[n_] := Denominator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]

a(n) = denominator(bernfrac(2*n)*(sum(k=1,2*n-1,1/k)^2 - sum(k=1,2*n-1,1/k^2))/(2*n)); \\ Michel Marcus, Sep 23 2015

a(n) = denominator(B_{2n}*(H^2_{2n-1}-H^(2)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.

a(n) = denominator(-Zeta(1 - 2*n)*(Psi(1,2*n) + (Psi(0,2*n) + gamma)^2 - (Pi^2)/6)), where gamma is Euler's gamma and Psi is the digamma function. - Peter Luschny, Apr 19 2018

cp's OEIS Frontend

A001620 Decimal expansion of Euler's constant (or the Euler-Mascheroni constant), gamma.

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A002657 Numerators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

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A002790 Denominators of Cauchy numbers of second type (= Bernoulli numbers B_n^{(n)}).

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A075266 Numerator of the coefficient of x^n in log(-log(1-x)/x).

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A262382 Numerators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

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A262383 Denominators of a semi-convergent series leading to the first Stieltjes constant gamma_1.

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