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A113092 Square table T, read by antidiagonals, where T(n,k) gives the number of n-th generation descendents of a node labeled (k) in the tree of 4-tournament sequences. 9
1, 0, 1, 0, 1, 1, 0, 4, 2, 1, 0, 46, 13, 3, 1, 0, 1504, 242, 27, 4, 1, 0, 146821, 13228, 693, 46, 5, 1, 0, 45236404, 2241527, 52812, 1504, 70, 6, 1, 0, 46002427696, 1237069018, 12628008, 146821, 2780, 99, 7, 1, 0, 159443238441379, 2305369985312, 9924266772 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

A 4-tournament sequence is an increasing sequence of positive integers (t_1,t_2,...) such that t_1 = p, t_i = p (mod 3) and t_{i+1} <= 4*t_i, where p>=1. This is the table of 4-tournament sequences when the starting node has label p = k for column k>=1.

LINKS

Table of n, a(n) for n=0..48.

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+3*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. Also, column k of T equals column 0 of the matrix k-th power of triangle A113095, which satisfies the matrix recurrence: A113095(n, k) = [A113095^4](n-1, k-1) + [A113095^4](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,...

0,4,13,27,46,70,99,133,172,216,265,...

0,46,242,693,1504,2780,4626,7147,10448,14634,...

0,1504,13228,52812,146821,330745,648999,1154923,1910782,...

0,146821,2241527,12628008,45236404,124626530,289031301,...

0,45236404,1237069018,9924266772,46002427696,155367674020,...

0,46002427696,2305369985312,26507035453923,159443238441379,...

0,159443238441379,14874520949557933,246323730279500082,...

PROG

(PARI) /* Generalized Cook-Kleber Recurrence */

{T(n, k, q=4)=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=4)=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]))); return((M^k)[n+1, 1])}

for(n=0, 10, for(k=0, 10, print1(T(n, k), ", ")); print(""))

CROSSREFS

Cf. A113095, A113096 (column 1), A113098 (column 2), A113100 (column 2); Tables: A093729 (2-tournaments), A113081 (3-tournaments), A113103 (5-tournaments); diagonals: A113093, A113094.

Sequence in context: A266861 A265435 A277004 * A033324 A266145 A266144

Adjacent sequences:  A113089 A113090 A113091 * A113093 A113094 A113095

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Oct 14 2005

STATUS

approved

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Last modified January 27 05:19 EST 2021. Contains 340451 sequences. (Running on oeis4.)