A155456 Write (1+1/x)*log(1+x) = Sum c(n)*x^n; then a(n) = (n+1)!*c(n).

-1, -1, 1, -2, 6, -24, 120, -720, 5040, -40320, 362880, -3628800, 39916800, -479001600, 6227020800, -87178291200, 1307674368000, -20922789888000, 355687428096000, -6402373705728000, 121645100408832000

Offset: 0

Author

Roger L. Bagula, Jan 22 2009

Keywords

• sign
• easy

Comments

Apart from initial terms and signs, identical to A000142.

a(n-1), n >= 0, is the negative of the alternating row sum of A048994 (Stirling1) with e.g.f. -1/(1+x). - Wolfdieter Lang, May 09 2017

Links

• P. W. Anderson, D. J. Thouless, E. Abrahams and D. S. Fisher, New method for a scaling theory of localization, Physical Review B, 1980.

Crossrefs

Cf. A048994.

Programs

Mathematica

p[x] = -(1 + 1/x)*Log[1 + x];
Table[ (n + 1)!*SeriesCoefficient[ Series[p[x], {x, 0, 30}], n], {n, 0, 30}]

Formula

G.f.: -1-x+x^2/(G(0)+x) where G(k)= 1 + (k+1)*x/(1 + x*(k+2)/G(k+1)); (continued fraction, 2-step). - Sergei N. Gladkovskii, Aug 14 2012

G.f.: conjecture: T(0)*x^2/(1+2*x) - 1 - x, where T(k) = 1 - x^2*(k+1)*(k+2)/(x^2*(k+1)*(k+2) - (1+2*x*(k+1))*(1+2*x*(k+2))/T(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Nov 19 2013

Extensions

Edited by N. J. A. Sloane, Jun 02 2009