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%I #15 Nov 11 2016 21:48:57
%S 1,0,0,20,35,448,3024,27480,268125,2905760,34402368,442140972,
%T 6128803135,91137168640,1447072631840,24433531297776,437138635330137,
%U 8260372499542080,164393521482487360,3436814164696775940
%N Fourth column (k=3) of triangle A134832 (circular succession numbers).
%C a(n) enumerates circular permutations of {1,2,...,n+3} with exactly three successor pairs (i,i+1). Due to cyclicity also (n+3,1) is a successor pair.
%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=3.
%H G. C. Greubel, <a href="/A135801/b135801.txt">Table of n, a(n) for n = 0..446</a>
%F a(n) = binomial(n+3,3)*A000757(n), n>=0.
%F E.g.f.: (d^3/dx^3) (x^3/3!)*(1-log(1-x))/e^x.
%e a(1)=0 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.
%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, 3], {n, 3, 10}] (* _G. C. Greubel_, Nov 10 2016 *)
%Y Cf. A134515 (column k=2), A135802 (column k=4).
%K nonn,easy
%O 0,4
%A _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008
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