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A135802 Fifth column (k=4) of triangle A134832 (circular succession numbers). 3
1, 0, 0, 35, 70, 1008, 7560, 75570, 804375, 9443720, 120408288, 1658028645, 24515212540, 387332966720, 6511826843280, 116059273664436, 2185693176650685, 43366955622595920, 904164368153680480 (list; graph; refs; listen; history; text; internal format)
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: A229357 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

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Last modified December 12 22:06 EST 2019. Contains 329963 sequences. (Running on oeis4.)