A001340 E.g.f.: 2*exp(x)/(1-x)^3.

2, 8, 38, 212, 1370, 10112, 84158, 780908, 8000882, 89763320, 1094915222, 14431179908, 204423631178, 3097603939952, 50001759773870, 856665220770332, 15526612798028258, 296825612428239848, 5969385443426556422, 125983618731675924020, 2784204907403441680442

Offset: 0

Author

N. J. A. Sloane and Simon Plouffe

Keywords

• nonn
• easy

Comments

a(n) = A001339 (n+1) - A001339 (n)..3-1=2, 11-3=8, 49-11=38... [Gary Detlefs, Jun 06 2010]

References

• N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n = 0..100
• E. Biondi, L. Divieti, G. Guardabassi, Counting paths, circuits, chains and cycles in graphs: A unified approach, Canad. J. Math. 22 1970 22-35.
• Philip Feinsilver and John McSorley, Zeons, Permanents, the Johnson Scheme, and Generalized Derangements, International Journal of Combinatorics, Volume 2011, Article ID 539030, 29 pages; doi:10.1155/2011/539030.

Programs

Mathematica

nn = 20; Range[0, nn]! CoefficientList[Series[2*Exp[x]/(1 - x)^3, {x, 0, nn}], x] (* T. D. Noe, Jun 28 2012 *)

Formula

a(n) = 2 * A082030(n).

a(n) = floor((n+1)*(n+1)!*e) - floor(n*n!*e) [Gary Detlefs, Jun 06 2010]

a(n) = {exp(1)*(n^2+n+1)*n!} for n>0, where {x} is the neareast integer, proposed by Simon Plouffe, March 1993.

G.f.: (1-x)/x/Q(0) -1/x, where Q(k)= 1 - x - x*(k+2)/(1 - x*(k+1)/Q(k+1)); (continued fraction). - Sergei N. Gladkovskii, Apr 22 2013

G.f.: W(0)/x - 1/x, where W(k) = 1 - x*(k+2)/( x*(k+3) - 1/(1 - x*(k+1)/( x*(k+1) - 1/W(k+1) ))); (continued fraction). - Sergei N. Gladkovskii, Aug 26 2013

Conjecture: a(n) +(-n-3)*a(n-1) +(n-1)*a(n-2)=0. - R. J. Mathar, May 03 2017

Extensions

Error in description corrected Jan 30 2008

More terms from N. J. A. Sloane, Jan 30 2008