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Search: id:A113095
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| A113095 |
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Triangle T, read by rows, that satisfies the recurrence: T(n,k) = [T^4](n-1,k-1) + [T^4](n-1,k) for n>k>=0, with T(n,n)=1 for n>=0, where T^4 is the matrix 4-th power of T. |
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12
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| 1, 1, 1, 4, 5, 1, 46, 66, 21, 1, 1504, 2398, 978, 85, 1, 146821, 255113, 122914, 14962, 341, 1, 45236404, 84425001, 46001193, 7046354, 235122, 1365, 1, 46002427696, 91159696960, 54661544301, 9933169553, 432627794, 3738738, 5461, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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Column 0 of the matrix power p, T^p, equals the number of 4-tournament sequences having initial term p (see A113092 for definitions).
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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^4], and for all integer p>=1: GF[T^p] = 1 + x*Sum_{j=1..p} GF[T^(p+3*j)] + x*y*GF[T^(4*p)].
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EXAMPLE
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Triangle T begins:
1;
1,1;
4,5,1;
46,66,21,1;
1504,2398,978,85,1;
146821,255113,122914,14962,341,1;
45236404,84425001,46001193,7046354,235122,1365,1; ...
Matrix third power T^3 (A113099) begins:
1;
3,1;
27,15,1;
693,513,63,1;
52812,47619,8289,255,1; ...
where column 0 equals A113100.
Matrix 4-th power T^4 (A113101) begins:
1;
4,1;
46,20,1;
1504,894,84,1;
146821,108292,14622,340,1;
45236404,39188597,6812596,233758,1364,1; ...
where adjacent sums in row n of T^4 forms row n+1 of T.
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PROGRAM
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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^4)[r-1, c-1])+(M^4)[r-1, c]))); return(M[n+1, k+1])}
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CROSSREFS
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Cf. A097710, A113084, A113106; A113092, A113096 (column 0), A113097 (T^2), A113099 (T^3), A113101 T^4).
Sequence in context: A109962 A102230 A110519 this_sequence A016715 A085548 A074459
Adjacent sequences: A113092 A113093 A113094 this_sequence A113096 A113097 A113098
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KEYWORD
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nonn,tabl
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Oct 14 2005
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