A135807 - OEIS

%I #16 Jul 01 2023 08:27:42

%S 1,0,0,220,715,16016,180180,2619760,39503750,642172960,11111964864,

%T 204016477080,3959206825210,80952590044480,1739019535313720,

%U 39150661649469744,921633956154372175,22640304292494917600

%N Tenth column (k=9) of triangle A134832 (circular succession numbers).

%C a(n) enumerates circular permutations of {1,2,...,n+9} with exactly nine successor pairs (i,i+1). Due to cyclicity also (n+9,1) is a successor pair.

%D Ch. A. Charalambides, Enumerative Combinatorics, Chapman & Hall/CRC, Boca Raton, Florida, 2002, p. 183, eq. (5.15), for k=9.

%H G. C. Greubel, <a href="/A135807/b135807.txt">Table of n, a(n) for n = 0..440</a>

%F a(n) = binomial(n+9,9)*A000757(n), n>=0.

%F E.g.f.: (d^9/dx^9) (x^9/9!)*(1-log(1-x))/e^x.

%e a(0)=1 because from the 9!/9 = 40320 circular permutations of n=9 elements only one, namely (1,2,3,4,5,6,7,8,9), has nine 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, 9], {n, 9, 25}] (* _G. C. Greubel_, Nov 10 2016 *)

%o (PARI) a(n)=((-1)^n + sum( k=0, n-1, (-1)^k * binomial( n, k) * (n - k - 1)!))*binomial(n+9,9) \\ _Charles R Greathouse IV_, Nov 10 2016

%Y Cf. A135806 (column k=8).

%K nonn,easy

%O 0,4

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