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A113095
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 4th power of T.
11
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
OFFSET
0,4
COMMENTS
Column 0 of the matrix power p, T^p, equals the number of 4-tournament sequences having initial term p (see A113092 for definitions).
FORMULA
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)].
EXAMPLE
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 4th 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.
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^4)[r-1, c-1])+(M^4)[r-1, c]))); return(M[n+1, k+1])}
CROSSREFS
Cf. A097710, A113084, A113106; A113092, A113096 (column 0), A113097 (T^2), A113099 (T^3), A113101 (T^4).
Sequence in context: A286796 A286718 A204579 * A157784 A274615 A258895
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Oct 14 2005
STATUS
approved