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%I #16 Nov 11 2016 21:49:33
%S 1,0,0,165,495,10296,108108,1473615,20913750,321086480,5263562304,
%T 91807414686,1696802925090,33116968654560,680485905122760,
%U 14681498118551154,331788224215573983,7837028408940548400
%N Ninth column (k=8) of triangle A134832 (circular succession numbers).
%C a(n) enumerates circular permutations of {1,2,...,n+8} with exactly eight successor pairs (i,i+1). Due to cyclicity also (n+8,1) is a successor pair.
%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=8.
%H G. C. Greubel, <a href="/A135806/b135806.txt">Table of n, a(n) for n = 0..441</a>
%F a(n) = binomial(n+8,8)*A000757(n), n>=0.
%F E.g.f.: (d^8/dx^8) (x^8/8!)*(1-log(1-x))/e^x.
%e a(0)=1 because from the 8!/8 = 5040 circular permutations of n=8 elements only one, namely (1,2,3,4,5,6,7,8), has eight 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, 8], {n, 8, 25}] (* _G. C. Greubel_, Nov 10 2016 *)
%Y Cf. A135805 (column k=7), A135807 (column k=9).
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jan 21 2008