OFFSET
0,8
COMMENTS
A 5-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 4) and t_{i+1} <= 5*t_i, where p>=1. This is the table of 5-tournament sequences when the starting node has label p = k for column k>=1.
LINKS
M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
FORMULA
For n>=k>0: T(n, k) = Sum_{j=1..k} T(n-1, k+4*j); else for k>n>0: T(n, k) = Sum_{j=1..n+1}(-1)^(j-1)*C(n+1, j)*T(n, k-j); with T(0, k)=1 for k>=0. Column k of T equals column 0 of the matrix k-th power of triangle A113106, which satisfies the matrix recurrence: A113106(n, k) = [A113106^5](n-1, k-1) + [A113106^5](n-1, k) for n>k>=0.
EXAMPLE
Table begins:
1,1,1,1,1,1,1,1,1,1,1,1,1,...
0,1,2,3,4,5,6,7,8,9,10,11,...
0,5,16,33,56,85,120,161,208,261,320,...
0,85,440,1251,2704,4985,8280,12775,18656,26109,...
0,4985,43600,173505,481376,1082905,2122800,3774785,6241600,...
0,1082905,16698560,94216515,337587520,930005021,2156566656,...
0,930005021,26098464448,210576669921,978162377600,...
0,3306859233805,172513149018752,2002383115518243,...
0,50220281721033905,4938593053649344000,82856383278525698433,...
PROG
(PARI) /* Generalized Cook-Kleber Recurrence */
{T(n, k, q=5)=if(n==0, 1, if(n<0||k<=0, 0, if(n==1, k, if(n>=k, sum(j=1, k, T(n-1, k+(q-1)*j)), sum(j=1, n+1, (-1)^(j-1)*binomial(n+1, j)*T(n, k-j))))))}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
(PARI) /* Matrix Power Recurrence (Paul D. Hanna) */
{T(n, k, q=5)=local(M=matrix(n+1, n+1)); for(r=1, n+1, for(c=1, r, M[r, c]=if(r==c, 1, if(c>1, (M^q)[r-1, c-1])+(M^q)[r-1, c]))); (M^k)[n+1, 1]}
for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 14 2005
STATUS
approved