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A117250 Triangle T, read by rows, where matrix power T^2 has powers of 2 in the secondary diagonal: [T^2](n+1,n) = 2^(n+1), with all 1's in the main diagonal and zeros elsewhere. 8
1, 1, 1, -1, 2, 1, 4, -4, 4, 1, -40, 32, -16, 8, 1, 896, -640, 256, -64, 16, 1, -43008, 28672, -10240, 2048, -256, 32, 1, 4325376, -2752512, 917504, -163840, 16384, -1024, 64, 1, -899678208, 553648128, -176160768, 29360128, -2621440, 131072, -4096, 128, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,5

COMMENTS

More generally, if a lower triangular matrix T to the power p is given by: [T^p](n,k) = C(r,n-k)*p^(n-k)*q^(n*(n-1)/2-k*(k-1)/2) then, for all m, [T^m](n,k) = [prod_{j=0..n-k-1}(m*r-p*j)]/(n-k)!*q^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1. This triangle results when m=1, p=2, q=2, r=1.

LINKS

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

FORMULA

T(n,k) = A117251(n-k)*2^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-2*j)]/(n-k)!*2^(n*(n-1)/2 - k*(k-1)/2) for n>k>=0, with T(n,n) = 1.

EXAMPLE

Triangle T begins:

1;

1,1;

-1,2,1;

4,-4,4,1;

-40,32,-16,8,1;

896,-640,256,-64,16,1;

-43008,28672,-10240,2048,-256,32,1;

4325376,-2752512,917504,-163840,16384,-1024,64,1;

-899678208,553648128,-176160768,29360128,-2621440,131072,-4096,128,1;

Matrix square T^2 has powers of 2 in the 2nd diagonal:

1;

2,1;

0,4,1;

0,0,8,1;

0,0,0,16,1;

0,0,0,0,32,1;

0,0,0,0,0,64,1; ...

PROG

(PARI) {T(n, k)=local(m=1, p=2, q=2, r=1); prod(j=0, n-k-1, m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

CROSSREFS

Cf. A117251 (column 0); variants: A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117260 (p=-1, q=2), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Sequence in context: A008741 A110316 A111975 * A296337 A308432 A136692

Adjacent sequences:  A117247 A117248 A117249 * A117251 A117252 A117253

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Mar 14 2006

STATUS

approved

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Last modified August 21 08:47 EDT 2019. Contains 326162 sequences. (Running on oeis4.)