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

0,4

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.

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

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.

Example

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.

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 *)

Crossrefs

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

Sequence in context: A286980 A254463 A030443 * A048452 A306935 A253418

Adjacent sequences: A135800 A135801 A135802 * A135804 A135805 A135806

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008

Status

approved