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.
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
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,...
Links
- M. Cook and M. Kleber, Tournament sequences and Meeussen sequences, Electronic J. Comb. 7 (2000), #R44.
Crossrefs
Programs
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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.
Comments