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

Author

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

Keywords

• nonn
• easy

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.

Examples

			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.
		

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

Crossrefs

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

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

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.