A135803 -id:A135803 - cp's OEIS frontend

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

Wolfdieter Lang, Jan 21 2008, Feb 22 2008

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.

			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.

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

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

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

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 *)

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.

A135804 Seventh column (k=6) of triangle A134832 (circular succession numbers).

Original entry on oeis.org

1, 0, 0, 84, 210, 3696, 33264, 392964, 4879875, 66106040, 963266304, 15032793048, 250055167908, 4415595820608, 82483140014880, 1624829831302104, 33659674920420549, 731455984834451184, 16636624374027720832

Offset: 0

Wolfdieter Lang, Jan 21 2008

a(n) enumerates circular permutations of {1,2,...,n+6} with exactly six successor pairs (i,i+1). Due to cyclicity also (n+6,1) is a successor pair.

			a(0)=1 because from the 6!/6 = 120 circular permutations of n=6 elements only one, namely (1,2,3,4,5,6), has six successors.

Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=6.

G. C. Greubel, Table of n, a(n) for n = 0..444

Cf. A135803 (column k=5), A135805 (column k=7).

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, 6], {n, 6, 25}] (* G. C. Greubel, Nov 10 2016 *)

a(n) = binomial(n+6,6)*A000757(n), n>=0.

E.g.f.: (d^6/dx^6) (x^6/6!)*(1-log(1-x))/e^x.

cp's OEIS Frontend

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

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