|
|
A262385
|
|
Denominators of a semi-convergent series leading to the second Stieltjes constant gamma_2.
|
|
6
|
|
|
1, 60, 336, 21600, 133056, 825552000, 89100, 11435424000, 483113030400, 101889627840000, 1471926193920, 42280119968486400, 3425059028160, 209827678712652000, 1184296360402995840, 163066081742403840000, 1749151741873536000, 20373357051590182072392960000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
gamma_2 = - 1/60 + 5/336 - 469/21600 + 6515/133056 - 131672123/825552000 + ..., see formulas (46)-(47) in the reference below.
|
|
LINKS
|
|
|
FORMULA
|
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
|
|
EXAMPLE
|
Denominators of 0/1, -1/60, 5/336, -469/21600, 6515/133056, -131672123/825552000, ...
|
|
MAPLE
|
a := n -> denom(-Zeta(1 - 2*n)*(Psi(1, 2*n) + (Psi(0, 2*n) + gamma)^2 - (Pi^2)/6)):
|
|
MATHEMATICA
|
a[n_] := Denominator[BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^2 - HarmonicNumber[2*n - 1, 2])/(2*n)]; Table[a[n], {n, 1, 20}]
|
|
PROG
|
(PARI) 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
|
|
CROSSREFS
|
Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384 (numerators of this series), A086280, A262386, A262387.
|
|
KEYWORD
|
nonn,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|