W. Lang, Nov 30 2007

a(n,k)=C(n,k) tabl head (triangle) for A134832 (nr. of circular permutations of {1,2,...,n} 
with exactly k successor pairs (i,i+1) ). 
Note that due to cyclicity also (n,1) is a successor pair. 


 n\k           0        1       2       3       4      5     6    7    8    9 ...


    0          1        0       0       0       0      0     0    0    0    0

    1          0        1       0       0       0      0     0    0    0    0

    2          0        0       1       0       0      0     0    0    0    0

    3          1        0       0       1       0      0     0    0    0    0

    4          1        4       0       0       1      0     0    0    0    0

    5          8        5      10       0       0      1     0    0    0    0

    6         36       48      15      20       0      0     1    0    0    0

    7        229      252     168      35      35      0     0    1    0    0

    8       1625     1832    1008     448      70     56     0    0    1    0

    9      13208    14625    8244    3024    1008    126    84    0    0    1
    .
    .
    .


 First (k=0) column:  A000757= [1,0,0,1,1,8,36,229,1625,13208,120288,1214673,13469897,...],
 circular permutations without any successor pair.

 Row sums give (n-1)!= A000142(n-1), n>=1.

 Alternating row sums give: A134833 = [1,-1,1,0,-2,12,-16,144,368,4768,...].


Because C(n,k)=binomial(n,k)*C(n-k,0), k>=1, this is a Scheffer triangle of the Appell type:
((1-ln(1-x))/e^x,x). See the e.g.f. of the first column (k=0) sequence A000757. 
I.e. the e.g.f. for column nr. k is ((1-ln(1-x))/e^x)*(x^k)/k!, k>=0, hence 
a(n,k)=binomial(n,k)*a(n-k,0), n>=0. 

The a-sequence for Appel type Sheffer triangles has always (e.)g.f 1.
The z-sequence for such triangles is always (1+1/g(x))/x if g is the e.g.f. of the first (k=0) column. 


###################################### e.o.f. ########################################################