cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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.

Original entry on oeis.org

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
Offset: 0

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Author

Paul D. Hanna, Oct 14 2005

Keywords

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.

Examples

			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,...
		

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.

Programs

  • 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(""))

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.