A194506 -id:A194506 - cp's OEIS frontend

A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

1, 1, 7, 17, 41, 731, 8563, 27719, 190073, 516149, 1013143139, 1519024289, 14108351869, 14399405173, 23142912688967, 83945247395407, 84894728616107, 3204549982389941, 262488267575333123, 9027726081126601799, 2026692221793223022131, 1375035304877251309001

Offset: 0

Paul Curtz, Aug 27 2011

Akiyama-Tanigawa from 1/n gives Bernoulli A164555(n)/A027642(n).
Reciprocally
1,   1/2,   5/12,     3/8, 251/720,    95/288, 19087/60480, 5257/17280,
1/2, 1/6,    1/8,  19/180,    3/32, 863/10080,    275/3456,
1/3, 1/12, 7/120,  17/360, 41/1008, 731/20160, 8563/259200,
1/4, 1/20, 1/30,   11/420, 89/4032,5849/302400,
1/5, 1/30, 3/140, 83/5040, 59/4320,
1/6, 1/42, 5/336,
1/7, 1/56,
1/8.
First row: A002208/A002209 or reduced A002657(n)/A091137(n) unsigned.
Second row: A002206(n+1)/A002689(n) unsigned. See A141417(n) and A174727(n).
Third row: a(n)/A194506(n).

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. A194506 (denominator).

read("transforms3") ;
L := [seq(1/n,n=1..20)] ;
L1 := AKIYAMATANIGAWAi(L) ;
L2 := AKIYATANI(L1) ;
L3 := AKIYATANI(L2) ;
apply(numer,%) ; # R. J. Mathar, Aug 27 2011
# second Maple program:
b:= proc (n, k) option remember;
      `if`(n=0, 1/(k+1), b(n-1, k) -b(n-1, k+1)/n)
    end:
a:= n-> numer(b(n, 2)):
seq(a(n), n=0..30);  # Alois P. Heinz, Aug 27 2011

a[n_, 0] := 1/(n+1); a[n_, m_] := a[n, m] = a[n, m-1] - a[n+1, m-1]/m; Table[a[2, m], {m, 0, 21}] // Numerator (* Jean-François Alcover, Aug 09 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_{0Vladimir 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)).

cp's OEIS Frontend

A193546 Numerator 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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cp's OEIS Frontend

A193546 Numerator of the third row of the inverse Akiyama-Tanigawa algorithm from 1/n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

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

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