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A113106
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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.
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13
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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
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OFFSET
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0,4
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COMMENTS
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Column 0 of the matrix power p, T^p, equals the number of 5-tournament sequences having initial term p (see A113103 for definitions).
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LINKS
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FORMULA
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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)].
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EXAMPLE
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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; ...
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.
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PROG
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(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])}
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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