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A109152 Triangle T, read by rows, that satisfies: T(n,k) = [T^2](n-1,k) for n>k+1>=1, with T(n,n) = 1 and T(n+1,n) = n+1 for n>=0, where T^2 is the matrix square of T. 7
1, 1, 1, 2, 2, 1, 6, 4, 3, 1, 22, 14, 6, 4, 1, 94, 56, 24, 8, 5, 1, 450, 262, 102, 36, 10, 6, 1, 2366, 1348, 516, 160, 50, 12, 7, 1, 13450, 7574, 2814, 868, 230, 66, 14, 8, 1, 81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1, 527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET
0,4
LINKS
FORMULA
T^(m+1) = SHIFT_UP(T^m - T^(m-1)) - D*T^(m-1) for all m where diagonal matrix D = [0, 1, 2, 3, ...] and SHIFT_UP shifts each column up 1 row.
Recurrence:
T(n,k) = Sum_{j=k..n-1} T(n-1, j) * T(j, k) for k=0..n-2 for n>1 with T(n+1,n) = n+1 and T(n,n) = 1 for n>=0. - Paul D. Hanna, Jan 13 2016
EXAMPLE
Triangle T begins:
1;
1, 1;
2, 2, 1;
6, 4, 3, 1;
22, 14, 6, 4, 1;
94, 56, 24, 8, 5, 1;
450, 262, 102, 36, 10, 6, 1;
2366, 1348, 516, 160, 50, 12, 7, 1;
13450, 7574, 2814, 868, 230, 66, 14, 8, 1;
81802, 45616, 16752, 4968, 1330, 312, 84, 16, 9, 1;
527826, 292442, 106026, 31148, 7930, 1914, 406, 104, 18, 10, 1; ...
Matrix square T^2 starts:
1;
2 1;
6 4 1;
22 14 6 1;
94 56 24 8 1;
450 262 102 36 10 1;
2366 1348 516 160 50 12 1;
13450 7574 2814 868 230 66 14 1;
81802 45616 16752 4968 1330 312 84 16 1;
527826 292442 106026 31148 7930 1914 406 104 18 1;
3590294 1978788 712260 205888 52130 11820 2632 512 126 20 1; ...
which equals SHIFT_UP(T) - D where
D is the diagonal matrix [0,1,2,3,...].
PROG
(PARI) {T(n, k) = local(M=matrix(n, n, r, c, if(r>=c, T(r-1, c-1)))); if(n<k||k<0, 0, if(n==k||n==0, 1, if(n==k+1, n, (M^2)[n, k+1])))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print(""))
(PARI) {T(n, k) = if(n==k, 1, if(n==k+1, n, sum(j=k, n-1, T(n-1, j) * T(j, k) )))}
for(n=0, 10, for(k=0, n, print1(T(n, k), ", ")); print("")) \\ Paul D. Hanna, Jan 13 2016
CROSSREFS
Cf. A109153 (column 0), A109154 (column 1), A109155 (column 2), A109156 (row sums).
Cf. A152391.
Sequence in context: A178886 A065052 A128308 * A130469 A106381 A064784
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jun 20 2005
STATUS
approved

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Last modified February 24 05:36 EST 2024. Contains 370291 sequences. (Running on oeis4.)