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A321943
Decimal expansion of Ni_1 = (1/2)*(gamma - log(2*Pi)) + 1, where gamma is Euler's constant (or the Euler-Mascheroni constant).
1
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
0,1
COMMENTS
This constant links Euler's constant and Pi to the values of the Riemann zeta function at positive integers (see formulas).
REFERENCES
D. Suryanarayana, Sums of Riemann zeta function, Math. Student, 42 (1974), 141-143.
LINKS
B. Candelpergher, Ramanujan summation of divergent series, HAL Id : hal-01150208; Lecture Notes in Math. Series (Springer), 2185, (2017), 93.
R. J. Singh, 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.
FORMULA
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)).
EXAMPLE
0.369669299246093688522926308635583575659682194332178386585...
MAPLE
Digits := 100; evalf((1/2)*(gamma-ln(2*Pi))+1);
MATHEMATICA
First[RealDigits[N[(1/2)*(EulerGamma-Log[2*Pi])+1, 100], 10]]
PROG
(PARI) (1/2)*(Euler-log(2*Pi))+1
(Python)
from mpmath import *
mp.dps = 100; mp.pretty = True
+(1/2)*(euler-log(2*pi))+1
CROSSREFS
Cf. A001620 (Euler's constant), A000796 (Pi).
Sequence in context: A151862 A067722 A049341 * A351189 A187082 A377030
KEYWORD
nonn,cons
AUTHOR
Stefano Spezia, Dec 12 2018
STATUS
approved