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

0,4

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.

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

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A215967 A029563 A145665 * A157090 A124409 A241614

Adjacent sequences: A135803 A135804 A135805 * A135807 A135808 A135809

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008

Status

approved