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A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.

%I A113106 #6 Mar 13 2015 22:31:56
%S A113106 1,1,1,5,6,1,85,115,31,1,4985,7420,2590,156,1,1082905,1744965,723370,
%T A113106 62090,781,1,930005021,1601759426,752616215,82390620,1532715,3906,1,
%U A113106 3306859233805,6024941167511,3117415999361,409321203715,10025307495
%N A113106 Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^5](n-1,k-1) + [T^5](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^5 is the matrix 5th power of T.
%C A113106 Column 0 of the matrix power p, T^p, equals the number of 5-tournament sequences having initial term p (see A113103 for definitions).
%F A113106 Let GF[T] denote the g.f. of triangular matrix T. Then GF[T] = 1 + x*(1+y)*GF[T^5] and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+4*j)] + x*y*GF[T^(5*p)].
%e A113106 Triangle begins:
%e A113106 1;
%e A113106 1,1;
%e A113106 5,6,1;
%e A113106 85,115,31,1;
%e A113106 4985,7420,2590,156,1;
%e A113106 1082905,1744965,723370,62090,781,1;
%e A113106 930005021,1601759426,752616215,82390620,1532715,3906,1;
%e A113106 Matrix 4th power T^4 (A113112) begins:
%e A113106 1;
%e A113106 4,1;
%e A113106 56,24,1;
%e A113106 2704,1576,124,1;
%e A113106 481376,346624,39376,624,1; ...
%e A113106 where column 0 equals A113113.
%e A113106 Matrix 5th power T^5 (A113114) begins:
%e A113106 1;
%e A113106 5,1;
%e A113106 85,30,1;
%e A113106 4985,2435,155,1;
%e A113106 1082905,662060,61310,780,1;
%e A113106 930005021,671754405,80861810,1528810,3905,1; ...
%e A113106 where adjacent sums in row n of T^5 forms row n+1 of T.
%o A113106 (PARI) {T(n,k)=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^5)[r-1,c-1])+(M^5)[r-1,c]))); return(M[n+1,k+1])}
%Y A113106 Cf. A097710, A113084, A113095; A113103, A113107 (column 0), A113108 (T^2), A113110 (T^3), A113112 (T^4), A113112 (T^5).
%K A113106 nonn,tabl
%O A113106 0,4
%A A113106 _Paul D. Hanna_, Oct 14 2005