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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.
13
1, 1, 1, 5, 6, 1, 85, 115, 31, 1, 4985, 7420, 2590, 156, 1, 1082905, 1744965, 723370, 62090, 781, 1, 930005021, 1601759426, 752616215, 82390620, 1532715, 3906, 1, 3306859233805, 6024941167511, 3117415999361, 409321203715, 10025307495
OFFSET
0,4
COMMENTS
Column 0 of the matrix power p, T^p, equals the number of 5-tournament sequences having initial term p (see A113103 for definitions).
FORMULA
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)].
EXAMPLE
Triangle begins:
1;
1,1;
5,6,1;
85,115,31,1;
4985,7420,2590,156,1;
1082905,1744965,723370,62090,781,1;
930005021,1601759426,752616215,82390620,1532715,3906,1;
Matrix 4th power T^4 (A113112) begins:
1;
4,1;
56,24,1;
2704,1576,124,1;
481376,346624,39376,624,1; ...
where column 0 equals A113113.
Matrix 5th power T^5 (A113114) begins:
1;
5,1;
85,30,1;
4985,2435,155,1;
1082905,662060,61310,780,1;
930005021,671754405,80861810,1528810,3905,1; ...
where adjacent sums in row n of T^5 forms row n+1 of T.
PROG
(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])}
CROSSREFS
Cf. A097710, A113084, A113095; A113103, A113107 (column 0), A113108 (T^2), A113110 (T^3), A113112 (T^4), A113112 (T^5).
Sequence in context: A181612 A216808 A293107 * A171273 A373396 A157832
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 14 2005
STATUS
approved