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
KEYWORD
nonn,easy
AUTHOR
Wolfdieter Lang, Jan 21 2008, Feb 22 2008
STATUS
approved