|
|
A262387
|
|
Denominators of a semi-convergent series leading to the third Stieltjes constant gamma_3.
|
|
8
|
|
|
1, 120, 1008, 28800, 49896, 101088000, 5702400, 12350257920000, 43480172736000, 7075668600000, 206069667148800, 5919216795588096000, 581222138112000, 8460252005694128640000, 18991807088644406016000, 1150594272774401495040000, 33940540399314092544000, 9737059611553100811150566400000, 1290633707289706940160000, 1263402804161736165764268432000000
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
1,2
|
|
COMMENTS
|
gamma_3 = + 1/120 - 17/1008 + 967/28800 - 4523/49896 + 33735311/101088000 - ..., see formulas (46)-(47) in the reference below.
|
|
LINKS
|
|
|
FORMULA
|
a(n) = denominator(-B_{2n}*(H^3_{2n-1}-3*H_{2n-1}*H^(2)_{2n-1}+2*H^(3)_{2n-1})/(2n)), where B_n, H_n and H^(k)_n are Bernoulli, harmonic and generalized harmonic numbers respectively.
|
|
EXAMPLE
|
Denominators of -0/1, 1/120, -17/1008, 967/28800, -4523/49896, 33735311/101088000, ...
|
|
MATHEMATICA
|
a[n_] := Denominator[-BernoulliB[2*n]*(HarmonicNumber[2*n - 1]^3 - 3*HarmonicNumber[2*n - 1]*HarmonicNumber[2*n - 1, 2] + 2*HarmonicNumber[2*n - 1, 3])/(2*n)]; Table[a[n], {n, 1, 20}]
|
|
PROG
|
(PARI) a(n) = denominator(-bernfrac(2*n)*(sum(k=1, 2*n-1, 1/k)^3 -3*sum(k=1, 2*n-1, 1/k)*sum(k=1, 2*n-1, 1/k^2) + 2*sum(k=1, 2*n-1, 1/k^3))/(2*n));
|
|
CROSSREFS
|
Cf. A001620, A002206, A195189, A075266, A262235, A001067, A006953, A082633, A262382, A262383, A086279, A262384, A262385, A086280, A262386 (numerators of this series).
|
|
KEYWORD
|
nonn,frac
|
|
AUTHOR
|
|
|
STATUS
|
approved
|
|
|
|