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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 (list; table; graph; refs; listen; history; text; internal format)
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.

LINKS

Table of n, a(n) for n=0..65.

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

Cf. A091352, A091353, A091354, A104445.

Sequence in context: A101494 A125781 A091150 * A058730 A112705 A070895

Adjacent sequences:  A091348 A091349 A091350 * A091352 A091353 A091354

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Jan 02 2004

STATUS

approved

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Last modified March 30 19:49 EDT 2020. Contains 333127 sequences. (Running on oeis4.)