A009575 E.g.f. sinh(log(1+x))*log(1+x).

0, 0, 2, -6, 26, -140, 894, -6594, 55152, -515808, 5334480, -60453360, 745035840, -9921337920, 141972324480, -2172695011200, 35410730342400, -612356632473600, 11198955424204800, -215960417122867200, 4379675205233664000

Offset: 0

Author

R. H. Hardin

Keywords

• sign
• easy

Comments

abs(a(n)) is the number of cycles in all permutations of n elements with an even number of cycles. - Vladeta Jovovic, Sep 06 2007

Examples

			a(n) = ((-1)^n*(n!*h(n) + (n-2)!))/2, n > 1, where h(n) = Sum_{k=1..n} 1/k. - _Gary Detlefs_, Aug 31 2011
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..448

Crossrefs

Cf. A000254.

Programs

Mathematica

nn=30; Range[0, nn]! CoefficientList[Series[Sinh[Log[1 + x]]*Log[1 + x], {x, 0, nn}], x]
CoefficientList[Series[(x*(2 + x)*Log[1 + x])/(2*(1 + x)), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Jan 24 2015 *)

PARI

x='x+O('x^66);
concat([0,0],Vec(serlaplace(sinh(log(1+x))*log(1+x))))

Formula

E.g.f.: (1+x)*x/2*U(0) where U(k)= 1 - 1/((1+x)^2 - x*(1+x)^4*(k+1)/(x*(1+x)^2*(k+1) + (k+2)/U(k+1))) ; (continued fraction, 3rd kind, 3-step). - Sergei N. Gladkovskii, Sep 25 2012

a(n) ~ n! * (-1)^n * log(n) / 2 * (1 + gamma/log(n)), where gamma is the Euler-Mascheroni constant (A001620). - Vaclav Kotesovec, Jan 24 2015

Extensions

Extended with signs by Olivier Gérard, Mar 15 1997