A262858 -id:A262858 - cp's OEIS frontend

A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

1, 43, 20431, 2150797323119, 9020112358835722225404403, 51551916515442115079024221439308876243677598340510141

Offset: 1

Iaroslav V. Blagouchine, Oct 03 2015

gamma = 1 - 1/12 - 43/420 - 20431/240240 - 2150797323119/36100888223400 - ..., see formula (36) in the reference below.

			Numerators of 1/12, 43/420, 20431/240240, 2150797323119/36100888223400, ...

G. C. Greubel, Table of n, a(n) for n = 1..10
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. A075266, A075267, A001620, A195189, A002657, A002790, A262235, A075266, A006953, A001067, A262858 (denominators of this series).

List(List([1..6],n->n*Sum([2^n+1..2^(n+1)],k->(-1)^(k+1)/k)),NumeratorRat); # Muniru A Asiru, Oct 29 2018

[Numerator(n*(&+[(-1)^(k+1)/k: k in [2^n+1..2^(n+1)]])): n in [1..6]]; // G. C. Greubel, Oct 28 2018

a[n_] := Numerator[n*Sum[(-1)^(k + 1)/k, {k, 2^n + 1, 2^(n + 1)}]]; Table[a[n], {n, 1, 8}]

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

a(n) = n * Sum_{k = 2^n + 1 .. 2^(n + 1)} (-1)^(k + 1)/k.

A302120 Absolute value of the numerators of a series converging to Euler's constant.

Original entry on oeis.org

3, 11, 1, 311, 5, 7291, 243, 14462317, 3364621, 3337014731, 3155743303, 65528247068741, 2627553901, 1439156737843967, 2213381206625, 21757704362231905789, 2627003970197650333, 64925181492079668050329, 523317843775891637, 161371847993975070290712761, 78461950306245817433389909

Offset: 1

Iaroslav V. Blagouchine, Apr 01 2018

gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.

			Numerators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.

Cf. A302121 (denominators of this series), A262856, A262858.

[3] cat [Abs(Numerator( (1/2)*(-1)^(n+1)*(&+[StirlingFirst(n-1,k)*((-1/2)^(k+1) + 1)/(k+1): k in [1..n-1]])/Factorial(n) + (-1)^(n+1)*(&+[StirlingFirst(n,k)/(k+1): k in [1..n]])/(n*Factorial(n)) )): n in [2..30]]; // G. C. Greubel, Oct 29 2018

a:= proc(n) abs(numer((1/2)*(-1)^(n+1)*(add(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/(n)!+(-1)^(n+1)*(add(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*(n)!))) end proc: seq(a(n), n=1..23);

a[n_] := Numerator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1,l]*((-1/2)^(l+1) + 1)/(l+1),{l,0,n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1),{l,1,n}])/(n*n!)]; Table[Abs[a[n]], {n, 1, 24}]

a(n) = abs(numerator((1/2)*(-1)^(n+1)*(sum(l=0,n-1,stirling(n-1,l)*((-1/2)^(l+1) + 1)/(l+1))) /(n!) + (-1)^(n+1)*(sum(l=1,n,stirling(n,l)/(l+1)))/(n*n!)))

a(n) = abs(Numerators of ((1/2)*(-1)^(n+1)*(Sum_{l=0,n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1,n} S_1(n,l)/(l+1)))/(n*n!))), where S_1(x,y) are the signed Stirling numbers of the first kind.

A302121 Denominators of a series converging to Euler's constant.

Original entry on oeis.org

4, 96, 72, 46080, 1152, 2322432, 100352, 7431782400, 2090188800, 2452488192000, 2697737011200, 64274810535936000, 2923954176000, 1799694695006208000, 3085190905724928, 33566877054287216640000, 4458100858772520960000, 120538655501945394954240000, 1057781497894797312000

Offset: 1

Iaroslav V. Blagouchine, Apr 01 2018

gamma = 3/4 - 11/96 - 1/72 - 311/46080 - 5/1152 - 7291/2322432 - ..., see formula (104) in the reference below.

			Denominators of 3/4, -11/96, -1/72, -311/46080, -5/1152, -7291/2322432, ...

G. C. Greubel, Table of n, a(n) for n = 1..250
Ia. V. Blagouchine, Three Notes on Ser's and Hasse's Representations for the Zeta-functions. INTEGERS, Electronic Journal of Combinatorial Number Theory, vol. 18A, Article #A3, pp. 1-45, 2018. arXiv:1606.02044 [math.NT], 2016.

Cf. A302120 (numerators of this series), A262856, A262858.

a := proc (n) options operator, arrow; denum((1/2)*(-1)^(n+1)*(sum(Stirling1(n-1, l)*((-1/2)^(l+1)+1)/(l+1), l = 0 .. n-1))/factorial(n)+(-1)^(n+1)*(sum(Stirling1(n, l)/(l+1), l = 1 .. n))/(n*factorial(n))) end proc

a[n_] := Denominator[(1/2)*(-1)^(n+1)*(Sum[StirlingS1[n-1,l]*((-1/2)^(l+1) + 1)/(l+1),{l,0,n-1}])/(n!) + (-1)^(n+1)*(Sum[StirlingS1[n, l]/(l+1),{l,1,n}])/(n*n!)]; Table[a[n], {n, 1, 24}]

a(n) = denominator((1/2)*(-1)^(n+1)*(sum(l=0,n-1,stirling(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(sum(l=1,n,stirling(n,l)/(l+1)))/(n*n!))

a(n) = Denominators of ((1/2)*(-1)^(n+1)*(Sum_{l=0..n-1} (S_1(n-1,l)*((-1/2)^(l+1) + 1)/(l+1)))/(n!) + (-1)^(n+1)*(Sum_{l=1..n} S_1(n,l)/(l+1)))/(n*n!)), where S_1(x,y) are the signed Stirling numbers of the first kind.

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A262856 Numerators of the Nielsen-Jacobsthal series leading to Euler's constant.

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A302120 Absolute value of the numerators of a series converging to Euler's constant.

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Crossrefs

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Magma

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Mathematica

PARI

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A302121 Denominators of a series converging to Euler's constant.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula