A135802 Fifth column (k=4) of triangle A134832 (circular succession numbers).

1, 0, 0, 35, 70, 1008, 7560, 75570, 804375, 9443720, 120408288, 1658028645, 24515212540, 387332966720, 6511826843280, 116059273664436, 2185693176650685, 43366955622595920, 904164368153680480

Offset

0,4

Comments

a(n) enumerates circular permutations of {1,2,...,n+4} with exactly four successor pairs (i,i+1). Due to cyclicity also (n+4,1) is a successor pair.

References

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=4.

Links

G. C. Greubel, Table of n, a(n) for n = 0..445

Formula

a(n) = binomial(n+4,4)*A000757(n), n>=0.

E.g.f.: (d^4/dx^4) (x^4/4!)*(1-log(1-x))/e^x.

Example

a(0)=1 because the 4!/4 = 6 circular permutations of n=4 elements (1,2,3,4), (1,4,3,2), (1,3,4,2),(1,2,4,3), (1,4,2,3) and (1,3,2,4) have 4,0,1,1,1 and 1 successor pair, respectively. Hence (1,2,3,4) is the only circular permutation with 4 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, 4], {n, 4, 25}] (* G. C. Greubel, Nov 10 2016 *)

Crossrefs

Cf. A135801 (column k=3), A135803 (column k=5).

Sequence in context: A376713 A254364 A146207 * A043390 A031481 A044137

Adjacent sequences: A135799 A135800 A135801 * A135803 A135804 A135805

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Status

approved