A134515 Third column (k=2) of triangle A134832 (circular succession numbers).

1, 0, 0, 10, 15, 168, 1008, 8244, 73125, 726440, 7939008, 94744494, 1225760627, 17088219120, 255365758560, 4072255216296, 69021889788969, 1239055874931312, 23484788783212480, 468656477004105810, 9821896865573503095

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Keywords

• nonn
• easy

Comments

a(n) enumerates circular permutations of {1,2,...,n+2} with exactly two successor pairs (i,i+1). Due to cyclicity also (n+2,1) is a successor pair.

Examples

			a(2)=0 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1,1 and 1 successor pair, respectively.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=2.

Links

• Bhadrachalam Chitturi and Krishnaveni K S, Adjacencies in Permutations, arXiv preprint arXiv:1601.04469 [cs.DM], 2016.

Crossrefs

Cf. A135799 (column k=1).

Formula

E.g.f.: (d^2/dx^2) (x^2/2!)*(1-log(1-x))/e^x.

a(n) = (((n+2)*(n+1))/2)*A000757(n), n>=0.