A135806 Ninth column (k=8) of triangle A134832 (circular succession numbers).

1, 0, 0, 165, 495, 10296, 108108, 1473615, 20913750, 321086480, 5263562304, 91807414686, 1696802925090, 33116968654560, 680485905122760, 14681498118551154, 331788224215573983, 7837028408940548400

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy

Comments

a(n) enumerates circular permutations of {1,2,...,n+8} with exactly eight successor pairs (i,i+1). Due to cyclicity also (n+8,1) is a successor pair.

Examples

			a(0)=1 because from the 8!/8 = 5040 circular permutations of n=8 elements only one, namely (1,2,3,4,5,6,7,8), has eight successors.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=8.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..441

Crossrefs

Cf. A135805 (column k=7), A135807 (column k=9).

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, 8], {n, 8, 25}] (* G. C. Greubel, Nov 10 2016 *)

Formula

a(n) = binomial(n+8,8)*A000757(n), n>=0.

E.g.f.: (d^8/dx^8) (x^8/8!)*(1-log(1-x))/e^x.