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A104445 Triangular matrix T, read by rows, that satisfies: SHIFT_LEFT_UP(T) = T^2 - T + I, or, equivalently: T(n+1,k+1) = [T^2](n,k) - T(n,k) + [T^0](n,k) for n>=k>=0, with T(0,0)=1. 5
1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 4, 3, 1, 1, 1, 9, 9, 4, 1, 1, 1, 24, 30, 16, 5, 1, 1, 1, 77, 115, 70, 25, 6, 1, 1, 1, 295, 510, 344, 135, 36, 7, 1, 1, 1, 1329, 2602, 1908, 805, 231, 49, 8, 1, 1, 1, 6934, 15133, 11904, 5325, 1616, 364, 64, 9, 1, 1, 1, 41351, 99367, 83028, 39001 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,8

COMMENTS

Surprisingly, SHIFT_UP(T) = A091351, or T(n+1,k) = A091351(n,k) for n>=k>=0, where column k of A091351 equals column 0 of A091351^(k+1) for k>=0.

LINKS

Table of n, a(n) for n=0..70.

FORMULA

T(n, k) = Sum_{j=0..n-k-1} T(n-k, j)*T(j+k, k-1) for n>k>0 with T(n, 0)=T(n, n)=1 (n>=0).

EXAMPLE

Rows begin:

1;

1,1;

1,1,1;

1,2,1,1;

1,4,3,1,1;

1,9,9,4,1,1;

1,24,30,16,5,1,1;

1,77,115,70,25,6,1,1;

1,295,510,344,135,36,7,1,1;

1,1329,2602,1908,805,231,49,8,1,1;

1,6934,15133,11904,5325,1616,364,64,9,1,1; ...

PROG

(PARI) T(n, k)=if(n<k || k<0, 0, if(n==k || k==0, 1, sum(j=0, n-k-1, T(n-k, j)*T(j+k, k-1))))

CROSSREFS

Cf. A091351, A104446 (matrix square); columns form: A091352, A091353, A091354.

Sequence in context: A327502 A220632 A125653 * A000189 A000190 A003557

Adjacent sequences:  A104442 A104443 A104444 * A104446 A104447 A104448

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 07 2005

STATUS

approved

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Last modified May 12 02:06 EDT 2021. Contains 343808 sequences. (Running on oeis4.)