A135801 Fourth column (k=3) of triangle A134832 (circular succession numbers).

1, 0, 0, 20, 35, 448, 3024, 27480, 268125, 2905760, 34402368, 442140972, 6128803135, 91137168640, 1447072631840, 24433531297776, 437138635330137, 8260372499542080, 164393521482487360, 3436814164696775940

Offset

0,4

Comments

a(n) enumerates circular permutations of {1,2,...,n+3} with exactly three successor pairs (i,i+1). Due to cyclicity also (n+3,1) is a successor pair.

References

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=3.

Links

G. C. Greubel, Table of n, a(n) for n = 0..446

Formula

a(n) = binomial(n+3,3)*A000757(n), n>=0.

E.g.f.: (d^3/dx^3) (x^3/3!)*(1-log(1-x))/e^x.

Example

a(1)=0 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.

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, 3], {n, 3, 10}] (* G. C. Greubel, Nov 10 2016 *)

Crossrefs

Cf. A134515 (column k=2), A135802 (column k=4).

Sequence in context: A287055 A326403 A335251 * A078210 A174628 A316098

Adjacent sequences: A135798 A135799 A135800 * A135802 A135803 A135804

Keyword

nonn,easy

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Status

approved