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%I #14 Nov 11 2016 21:49:10
%S 1,0,0,56,126,2016,16632,181368,2091375,26442416,361224864,5305691664,
%T 83351722636,1394398680192,24744942004464,464237094657744,
%U 9179911341932877,190814604739422048,4159156093506930208
%N Sixth column (k=5) of triangle A134832 (circular succession numbers).
%C a(n) enumerates circular permutations of {1,2,...,n+5} with exactly five successor pairs (i,i+1). Due to cyclicity also (n+5,1) is a successor pair.
%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=5.
%H G. C. Greubel, <a href="/A135803/b135803.txt">Table of n, a(n) for n = 0..444</a>
%F a(n) = binomial(n+5,5)*A000757(n), n>=0.
%F E.g.f.: (d^5/dx^5) (x^5/5!)*(1-log(1-x))/e^x.
%e a(0)=1 because from the 5!/5 = 24 circular permutations of n=5 elements only one, namely (1,2,3,4,5), has five successors.
%t 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, 5], {n, 5, 25}] (* _G. C. Greubel_, Nov 10 2016 *)
%Y Cf. A135802 (column k=4), A135804 (column k=6).
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jan 21 2008
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