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A091351
Triangle T, read by rows, such that T(n,k) equals the (n-k)-th row sum of T^k, where T^k is the k-th power of T as a lower triangular matrix.
21
1, 1, 1, 1, 2, 1, 1, 4, 3, 1, 1, 9, 9, 4, 1, 1, 24, 30, 16, 5, 1, 1, 77, 115, 70, 25, 6, 1, 1, 295, 510, 344, 135, 36, 7, 1, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 41351, 99367, 83028, 39001, 12381, 2919, 540, 81, 10, 1
OFFSET
0,5
COMMENTS
Since T(n,0)=1 for n>=0, then the k-th column of the lower triangular matrix T equals the leftmost column of T^(k+1) for k>=0.
FORMULA
T(n, k) = sum_{j=0..n-k} T(n-k, j)*T(j+k-1, k-1) for n>=k>0 with T(n, 0)=1 (n>=0).
Equals SHIFT_UP(A104445), or A104445(n+1, k) = T(n, k) for n>=k>=0, where triangular matrix X=A104445 satisfies: SHIFT_LEFT_UP(X) = X^2 - X + I.
EXAMPLE
T(7,3) = 344 = 1*1 + 9*3 + 9*9 + 4*30 + 1*115
= T(4,0)*T(2,2) +T(4,1)*T(3,2) +T(4,2)*T(4,2) +T(4,3)*T(5,2) +T(4,4)*T(6,2).
Rows begin:
{1},
{1,1},
{1,2,1},
{1,4,3,1},
{1,9,9,4,1},
{1,24,30,16,5,1},
{1,77,115,70,25,6,1},
{1,295,510,344,135,36,7,1},
{1,1329,2602,1908,805,231,49,8,1},
{1,6934,15133,11904,5325,1616,364,64,9,1},...
PROG
(PARI) T(n, k)=if(k>n || n<0 || k<0, 0, if(k==0 || k==n, 1, sum(j=0, n-k, T(n-k, j)*T(j+k-1, k-1)); ); )
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Paul D. Hanna, Jan 02 2004
STATUS
approved