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A126465 Triangle T, read by rows, where row n equals row (n-1) of matrix power T^(n(n+1)/2) concatenated with a trailing '1', for n>0, with T(0,0) = 1. 10
1, 1, 1, 3, 1, 1, 33, 6, 1, 1, 855, 105, 10, 1, 1, 40475, 3710, 255, 15, 1, 1, 3039204, 219625, 11935, 525, 21, 1, 1, 331630320, 19545316, 879571, 31584, 966, 28, 1, 1, 49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

COMMENTS

Amazingly, A126465 = A126450*A126445^-1 = A126454*A126450^-1 = A126457*A126454^-1; and also A126460 = A126445^-1*A126450 = A126450^-1*A126454 = A126454^-1*A126457.

LINKS

Table of n, a(n) for n=0..44.

EXAMPLE

Triangle T begins:

1,

1, 1,

3, 1, 1,

33, 6, 1, 1,

855, 105, 10, 1, 1,

40475, 3710, 255, 15, 1, 1,

3039204, 219625, 11935, 525, 21, 1, 1,

331630320, 19545316, 879571, 31584, 966, 28, 1, 1,

49563943161, 2437990653, 93365328, 2856819, 72786, 1638, 36, 1, 1, ...

Matrix cube T^3 begins:

1;

[3, 1]; <-- row 1 of T^3 + '1' = row 2 of T;

12, 3, 1; ...

Matrix power T^6 begins:

1;

6, 1;

[33, 6, 1]; <-- row 2 of T^6 + '1' = row 3 of T.

Matrix power T^10 begins:

1;

10, 1;

75, 10, 1;

[855, 105, 10, 1]; <-- row 3 of T^10 + '1' = row 4 of T.

Matrix power T^15 begins:

1;

15, 1;

150, 15, 1;

1895, 195, 15, 1;

[40475, 3710, 255, 15, 1]; <-- row 4 of T^15 + '1' = row 5 of T.

PROG

(PARI) {T(n, k)=local(M=matrix(n+1, n+1, r, c, if(r>=c, binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!, r-c))), N=matrix(n+1, n+1, r, c, if(r>=c, binomial((r-1)*r*(r+1)/3!-(c-1)*c*(c+1)/3!+1, r-c)))); (N*M^-1)[n+1, k+1]}

CROSSREFS

Columns: A126466, A126467, A126468; A126469 (row sums); A126460 (dual); A101479 (variant).

Sequence in context: A168242 A332540 A215750 * A077509 A156698 A083993

Adjacent sequences:  A126462 A126463 A126464 * A126466 A126467 A126468

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Dec 27 2006

STATUS

approved

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Last modified May 17 10:57 EDT 2021. Contains 343971 sequences. (Running on oeis4.)