A135805 Eighth column (k=7) of triangle A134832 (circular succession numbers).

1, 0, 0, 120, 330, 6336, 61776, 785928, 10456875, 151099520, 2339361024, 38655753552, 678721170036, 12615988058880, 247449420044640, 5106608041235184, 110596074738524661, 2507849090860975488

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy

Comments

a(n) enumerates circular permutations of {1,2,...,n+7} with exactly seven successor pairs (i,i+1). Due to cyclicity also (n+7,1) is a successor pair.

Examples

			a(0)=1 because from the 7!/7 = 720 circular permutations of n=7 elements only one, namely (1,2,3,4,5,6,7), has seven successors.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=7.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..443

Crossrefs

Cf. A135804 (column k=6), A135806 (column k=8).

Programs

Mathematica

f[n_] := (-1)^n + Sum[(-1)^k*n!/((n - k)*k!), {k, 0, n - 1}]; a[n_, n_] = 1; a[n_, 0] := f[n]; a[n_, k_] := a[n, k] = n/k*a[n - 1, k - 1]; Table[a[n, 7], {n, 7, 25}] (* G. C. Greubel, Nov 10 2016 *)

Formula

a(n) = binomial(n+7,7)*A000757(n), n>=0.

E.g.f.: (d^7/dx^7) (x^7/7!)*(1-log(1-x))/e^x.