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

0,4

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.

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

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A090391 A275083 A098114 * A327912 A118058 A269037

Adjacent sequences: A135802 A135803 A135804 * A135806 A135807 A135808

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008

Status

approved