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%I #15 Nov 11 2016 21:49:04
%S 1,0,0,35,70,1008,7560,75570,804375,9443720,120408288,1658028645,
%T 24515212540,387332966720,6511826843280,116059273664436,
%U 2185693176650685,43366955622595920,904164368153680480
%N Fifth column (k=4) of triangle A134832 (circular succession numbers).
%C 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.
%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=4.
%H G. C. Greubel, <a href="/A135802/b135802.txt">Table of n, a(n) for n = 0..445</a>
%F a(n) = binomial(n+4,4)*A000757(n), n>=0.
%F E.g.f.: (d^4/dx^4) (x^4/4!)*(1-log(1-x))/e^x.
%e 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.
%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, 4], {n, 4, 25}] (* _G. C. Greubel_, Nov 10 2016 *)
%Y Cf. A135801 (column k=3), A135803 (column k=5).
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008
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