A060746 Absolute value of numerator of non-Euler-constant term of Laurent expansion of Gamma function at s = -n.

0, 1, 3, 11, 25, 137, 49, 121, 761, 7129, 7381, 83711, 86021, 1145993, 1171733, 1195757, 2436559, 42142223, 14274301, 275295799, 11167027, 18858053, 6364399, 444316699, 269564591, 34052522467, 34395742267, 312536252003

Offset: 0

Author

Sen-Peng Eu, Apr 23 2001

Keywords

• nonn

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

It appears that, if we discard the first term and set a(0)=1, then a(n) = denominator of n!(h(n)/h(n+1)) where h(n) is the n-th harmonic number = Sum_{k=1..n} 1/k. - Gary Detlefs, Sep 09 2010

Examples

			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.
		

Formula

Conjecture: a(n) = lcm(Wolstenholme(n), n!)/n!, cf. A001008. - Vladeta Jovovic, May 20 2004

Conjecture: a(n) = numerator(harmonic(n)/(n-1)!) for n >= 1. - Peter Luschny, May 13 2023