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A117260 Triangle T, read by rows, where matrix inverse T^-1 has -2^n in the secondary diagonal: [T^-1](n+1,n) = -2^n, with all 1's in the main diagonal and zeros elsewhere. 8
1, 1, 1, 2, 2, 1, 8, 8, 4, 1, 64, 64, 32, 8, 1, 1024, 1024, 512, 128, 16, 1, 32768, 32768, 16384, 4096, 512, 32, 1, 2097152, 2097152, 1048576, 262144, 32768, 2048, 64, 1, 268435456, 268435456, 134217728, 33554432, 4194304, 262144, 8192, 128, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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=-1, q=2, r=1.

LINKS

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

FORMULA

T(n,k) = 2^(n*(n-1)/2 - k*(k-1)/2).

EXAMPLE

Triangle T begins:

1;

1,1;

2,2,1;

8,8,4,1;

64,64,32,8,1;

1024,1024,512,128,16,1;

32768,32768,16384,4096,512,32,1;

2097152,2097152,1048576,262144,32768,2048,64,1;

268435456,268435456,134217728,33554432,4194304,262144,8192,128,1;

Matrix inverse T^-1 has -2^n in the 2nd diagonal:

1,

-1,1,

0,-2,1,

0,0,-4,1,

0,0,0,-8,1,

0,0,0,0,-16,1,

0,0,0,0,0,-32,1, ...

MATHEMATICA

Flatten[Table[2^((n(n-1))/2-(k(k-1))/2), {n, 0, 10}, {k, 0, n}]] (* Harvey P. Dale, Sep 19 2013 *)

PROG

(PARI) {T(n, k)=local(m=1, p=-1, 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. A006125 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), A117254 (p=q=4), A117256 (p=q=5), A117258 (p=2, q=4), A117262 (p=-1, q=3), A117265 (p=-2, q=2).

Sequence in context: A198569 A135080 A238182 * A077944 A077992 A126586

Adjacent sequences:  A117257 A117258 A117259 * A117261 A117262 A117263

KEYWORD

nonn,tabl

AUTHOR

Paul D. Hanna, Mar 14 2006

STATUS

approved

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Last modified June 6 03:19 EDT 2020. Contains 334858 sequences. (Running on oeis4.)