A135807 Tenth column (k=9) of triangle A134832 (circular succession numbers).

1, 0, 0, 220, 715, 16016, 180180, 2619760, 39503750, 642172960, 11111964864, 204016477080, 3959206825210, 80952590044480, 1739019535313720, 39150661649469744, 921633956154372175, 22640304292494917600

Offset

0,4

Comments

a(n) enumerates circular permutations of {1,2,...,n+9} with exactly nine successor pairs (i,i+1). Due to cyclicity also (n+9,1) is a successor pair.

References

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=9.

Links

G. C. Greubel, Table of n, a(n) for n = 0..440

Formula

a(n) = binomial(n+9,9)*A000757(n), n>=0.

E.g.f.: (d^9/dx^9) (x^9/9!)*(1-log(1-x))/e^x.

Example

a(0)=1 because from the 9!/9 = 40320 circular permutations of n=9 elements only one, namely (1,2,3,4,5,6,7,8,9), has nine 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, 9], {n, 9, 25}] (* G. C. Greubel, Nov 10 2016 *)

Prog

(PARI) a(n)=((-1)^n + sum( k=0, n-1, (-1)^k * binomial( n, k) * (n - k - 1)!))*binomial(n+9, 9) \\ Charles R Greathouse IV, Nov 10 2016

Crossrefs

Cf. A135806 (column k=8).

Sequence in context: A184543 A234558 A211816 * A258545 A257361 A258538

Adjacent sequences: A135804 A135805 A135806 * A135808 A135809 A135810

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Status

approved