OFFSET
0,3
COMMENTS
If you start with log(z) and integrate it n times in succession, then you get z^n*log(z)/n! - K(n)*z^n where K(1)=1, K(2)=3/4, K(3)=11/36, K(4)=25/288, K(5)=137/7200, K(6)=49/14400, etc. - Warren D. Smith, Jan 01 2006
The coefficients in the expansion of the n-th integral of log(x)^m were discussed in Loeb & Rota, 1989 (briefly overviewed in page 12 of Dan Rus, 2025), with Warren D. Smith's comment being the m=1 special case. - Natalia L. Skirrow, Jun 07 2026
LINKS
Daniel E Loeb and Gian-Carlo Rota, Formal power series of logarithmic type, Advances in Mathematics, 1989, Vol. 75, Iss. 1, pp. 1-118.
Mircea Dan Rus, 2025, Yet another note on notation, arXiv:2501.08762 [math.HO], 2025.
Natalia L. Skirrow, more on Faulhabernoulli and Gregory § more log-gamma series things.
Natalia L. Skirrow, above the Stirling triangles § recursive integrals of powers of logs.
FORMULA
Conjecture: a(n) = lcm(Wolstenholme(n), n!)/n!, cf. A007406. - Vladeta Jovovic, May 20 2004
Equivalent conjecture: a(n) = numerator(harmonic(n)/(n-1)!) for n >= 1. - Peter Luschny, May 13 2023
Conjecture: for n>=2, a(n) = numerator of H(n)/((n-1)!*H(n-1)) where H(n) is the n-th harmonic number = Sum_{k=1..n} 1/k. - Gary Detlefs, Sep 09 2010
From Natalia L. Skirrow, Jun 07 2026: (Start)
Both conjectures are true.
Gamma(s+m) = Gamma(m)*(1 + digamma(m)*s + O(s^2)), and digamma(m) = H(m-1) - gamma.
Via Euler's reflection formula,
Gamma(s-n) = (1/Gamma(n+1-s)) / (sin(Pi*(s-n))/Pi)
= ((1 + digamma(n+1)*s + O(s^2))/n!) / ((-1)^n*(s + O(s^3))),
whose constant term is (-1)^n*digamma(n+1)/n!, whose rational part is (-1)^n*H(n)/n!, so a(n) = num(H(n)/n!) (Vladeta Jovovic's).
Let num(x) = numerator of x. Writing H(n) = p/q with p,q coprime, H(n)/H(n-1) = p*n/(p*n-q), so num(H(n)/H(n-1)) / num(H(n)) = n/gcd(q,p*n) = n/gcd(q,n) = num(n*H(n)) / num(H(n)) = A120263(n). Therefore Gary Detlefs's num(H(n)/((n-1)!*H(n-1))) is num(n*H(n)/(n-1)!), which equals a(n)'s definition since gcd(denominator(H(n)),n) = n.
(End)
EXAMPLE
series(Gamma(s), s=-4,1 ) = series(1/24*(s+4)^(-1)+(25/288-1/24*gamma)+O((s+4)),s=-4,1). Hence a(4)=25.
series(Gamma(s), s=-5,1 ) = series(-1/120*(s+5)^(-1)+(-137/7200+1/120*gamma)+O((s+5)),s=-5,1). Hence a(5)=137.
PROG
(Python)
from math import gcd
def a_gen():
yield 0
yield 1
a0 = 0; a1 = 1; n = 2
f = 4; z = 1; m = 1; k = 3
while True:
s = -k * a1 - m * a0
a0, a1 = a1, s
m = n * n; f *= m;
k += 2; n += 1; z = -z
yield z * s // gcd(s, f)
a = a_gen(); print([next(a) for _ in range(31)]) # Peter Luschny, Jun 08 2026
CROSSREFS
KEYWORD
nonn,frac,changed
AUTHOR
Sen-Peng Eu, Apr 23 2001
STATUS
approved