A135803 Sixth column (k=5) of triangle A134832 (circular succession numbers).

1, 0, 0, 56, 126, 2016, 16632, 181368, 2091375, 26442416, 361224864, 5305691664, 83351722636, 1394398680192, 24744942004464, 464237094657744, 9179911341932877, 190814604739422048, 4159156093506930208

Offset: 0

Author

Wolfdieter Lang, Jan 21 2008

Keywords

• nonn
• easy

Comments

a(n) enumerates circular permutations of {1,2,...,n+5} with exactly five successor pairs (i,i+1). Due to cyclicity also (n+5,1) is a successor pair.

Examples

			a(0)=1 because from the 5!/5 = 24 circular permutations of n=5 elements only one, namely (1,2,3,4,5), has five successors.
		

References

• Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=5.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..444

Crossrefs

Cf. A135802 (column k=4), A135804 (column k=6).

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, 5], {n, 5, 25}] (* G. C. Greubel, Nov 10 2016 *)

Formula

a(n) = binomial(n+5,5)*A000757(n), n>=0.

E.g.f.: (d^5/dx^5) (x^5/5!)*(1-log(1-x))/e^x.