cp's OEIS Frontend

A113077 Column 3 of square table A093729; a(n) gives the number of n-th generation descendents of a node labeled (3) in the tree of tournament sequences, for n>=0.

%I A113077 #10 Dec 13 2021 02:03:27
%S A113077 1,3,15,123,1656,36987,1391106,89574978,10036638270,1986129275673,
%T A113077 703168200003336,450303519404234922,526421174510139860241,
%U A113077 1132076561237754405471033,4507472672071759672232970720
%N A113077 Column 3 of square table A093729; a(n) gives the number of n-th generation descendents of a node labeled (3) in the tree of tournament sequences, for n>=0.
%C A113077 Also equals column 0 of the matrix cube of triangle A097710, which satisfies the matrix recurrence: A097710(n,k) = [A097710^2](n-1,k-1) + [A097710^2](n-1,k) for n>k>=0.
%H A113077 M. Cook and M. Kleber, <a href="https://doi.org/10.37236/1522">Tournament sequences and Meeussen sequences</a>, Electronic J. Comb. 7 (2000), #R44.
%e A113077 The tree of tournament sequences of descendents of a node labeled (3) begins:
%e A113077 [3]; generation 1: 3->[4,5,6]; generation 2: 4->[5,6,7,8],
%e A113077 5->[6,7,8,9,10], 6->[7,8,9,10,11,12]; ...
%e A113077 Then a(n) gives the number of nodes in generation n.
%e A113077 Also, a(n+1) = sum of labels of nodes in generation n.
%o A113077 (PARI) {a(n,q=2)=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^3)[n+1,1])}
%Y A113077 Cf. A113078, A113079.
%K A113077 nonn
%O A113077 0,2
%A A113077 _Paul D. Hanna_, Oct 14 2005