A193546 -id:A193546 - cp's OEIS frontend

A194506 Denominator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

3, 12, 120, 360, 1008, 20160, 259200, 907200, 6652800, 19160064, 39626496000, 62270208000, 603542016000, 640493568000, 1067062284288000, 4001483566080000, 4174096582656000, 162193467211776000, 13651830701752320000, 481714597618974720000

Offset: 0

Paul Curtz, Aug 27 2011

For the numerator sequence and detailed information see A193546.

Alois P. Heinz, Table of n, a(n) for n = 0..200
Iaroslav V. Blagouchine, Three notes on Ser's and Hasse's representation for the zeta-functions, Integers (2018) 18A, Article #A3.

Cf. A193546 (numerator).

a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; a[n_] := a[2, n]; Table[a[n] , {n, 0, 19}] // Denominator (* Jean-François Alcover, Sep 18 2012 *)
Numerator@Table[(-1)^n (n + 1) Integrate[FunctionExpand[x Binomial[x, n + 1]], {x, 0, 1}], {n, 0, 20}] (* Vladimir Reshetnikov, Feb 01 2017 *)

a(n)/A194506(n) = (-1)^n * (n+1) * Integral_{x=0..1} x*binomial(x,n+1). - Vladimir Reshetnikov, Feb 01 2017

A321943 Decimal expansion of Ni_1 = (1/2)(gamma - log(2Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).

Original entry on oeis.org

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

Offset: 0

Stefano Spezia, Dec 12 2018

This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).

			0.369669299246093688522926308635583575659682194332178386585...

D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.

Stefano Spezia, Table of n, a(n) for n = 0..10000
B. Candelpergher, Ramanujan summation of divergent series, HAL Id : hal-01150208; Lecture Notes in Math. Series (Springer), 2185, (2017), 93.
Marc-Antoine Coppo, A note on some alternating series involving zeta and multiple zeta values, Journal of Mathematical Analysis and Applications Volume 475, Issue 2, 15 July 2019, Pages 1831-1841; Preprint, , 2018.
Michael I. Shamos, A catalog of the real numbers, (2007). See p. 378.
R. J. Singh and V. P. Verma, Some series involving Riemann zeta function, Yokohama Math. J. 31 (1983), 1-4.
H. M. Srivastava, Sums of certain series of the Riemann zeta function, J. Math. Anal. App. 134 (1988), 129-140.

Cf. A001620 (Euler's constant), A000796 (Pi).

Cf. A193546, A000290, A194506, A131688.

Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);

First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]

PARI
```
(1/2)*(Euler-log(2*Pi))+1
```

from mpmath import *
mp.dps = 100; mp.pretty = True
+(1/2)*(euler-log(2*pi))+1

Ni_1 = Sum_{k>=2} (-1)^k*zeta(k)/(k+1).
Ni_1 = Sum_{n>0} (Integral_{x=0..1} x^2*(1-x)_{n-1} dx)/(n*n!), where (z)_n = z*(z+1)*(z+2)*...*(z+n-1) is the Pochhammer symbol.
Ni_1 = Sum_{n>=0} A193546(n)/(A000290(n + 1)*A194506(n)).

A232853 Repeat n+1 times A091137(n).

Original entry on oeis.org

1, 2, 2, 12, 12, 12, 24, 24, 24, 24, 720, 720, 720, 720, 720, 1440, 1440, 1440, 1440, 1440, 1440, 60480, 60480, 60480, 60480, 60480, 60480, 60480, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 120960, 3628800

Offset: 0

Paul Curtz, Dec 01 2013

A002657(n) and A091137(n) are linked to the Bernoulli numbers B n.
Unreduced differences table of A002657(n)/A091137(n):
1,            1/2,     5/12,    9/24,  251/720, 475/1440,...
-1/2,       -1/12,    -1/24, -19/720, -27/1440,... =-A141417(n+1)/A091137(n+1),
5/12,        1/24,   11/720, 11/1440,...
-9/24,    -19/720, -11/1440,...
251/720,  27/1440,...
-475/1440,... etc.
This is an autosequence of the second kind: its inverse binomial transform is the signed sequence with the main diagonal double of the first upper diagonal.
a(n) is the denominators written by antidiagonals.

			1,
2,   2,
12, 12, 12,
24, 24, 24, 24, etc.

P. Curtz, Intégration numérique des systèmes différentiels à conditions initiales. Centre de Calcul Scientifique de l'Armement, Arcueil 1969. Pages 36 and 56.

Cf. A195287, A002208/A002209 (reduced autosequence), A193546, A174727, A165313.

Repeat n+1 times A091137(n). Triangle.

cp's OEIS Frontend

A194506 Denominator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

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A321943 Decimal expansion of Ni_1 = (1/2)(gamma - log(2Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).

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A232853 Repeat n+1 times A091137(n).

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cp's OEIS Frontend

A194506 Denominator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

A321943 Decimal expansion of Ni_1 = (1/2)*(gamma - log(2*Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Python

Formula

A232853 Repeat n+1 times A091137(n).

Views

Author

Keywords

Comments

Examples

References

Crossrefs

Formula

A321943 Decimal expansion of Ni_1 = (1/2)(gamma - log(2Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).