%I #27 Aug 20 2017 11:53:46

%S 1,0,0,4,5,48,252,1832,14625,132080,1323168,14576076,175108661,

%T 2278429216,31920719820,479088848976,7669098865441,130426934203296,

%U 2348478878321248,44633950190867220,892899715052136645

%N Second column (k=1) of triangle A134832 (circular succession numbers).

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

%C The o.g.f. of this sequence seems to be the product of the o.g.f. for A000166 (derangements) by the fraction (1+2*x)/(1+x)^2 = 1 - x^2+ 2*x^3 - 3*x^4 + ... = 1 + sum( (-1)^i i x^(i+1), i=0..infinity) - _Thomas Baruchel_, Jan 08 2016

%C This correspond to the following transform: a(n) = b(n) - sum((-1)^(n + i) (n - i - 1)*b(i), (i=0..n-2)) - _Olivier Gérard_, Mar 05 2016

%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=1.

%H G. C. Greubel, <a href="/A135799/b135799.txt">Table of n, a(n) for n = 0..448</a>

%H Bhadrachalam Chitturi and Krishnaveni K S, <a href="https://arxiv.org/abs/1601.04469">Adjacencies in Permutations</a>, arXiv preprint arXiv:1601.04469 [cs.DM], 2016.

%F a(n) = (n+1)*A000757(n), n>=0.

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

%F O.g.f.: see comment section.

%e a(3)=4 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(s), 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, 1], {n, 21}] (* _Michael De Vlieger_, Jan 09 2016, after _Jean-François Alcover_ at A134832 *)

%Y Cf. A000757 (k=0 column), A134515 (k=2 column).

%K nonn,easy

%O 0,4

%A _Wolfdieter Lang_, Jan 21 2008, Feb 22 2008