login
The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A117252 Triangle T, read by rows, where matrix power T^3 has powers of 3 in the secondary diagonal: [T^3](n+1,n) = 3^(n+1), with all 1's in the main diagonal and zeros elsewhere. 8
1, 1, 1, -3, 3, 1, 45, -27, 9, 1, -2430, 1215, -243, 27, 1, 433026, -196830, 32805, -2187, 81, 1, -245525742, 105225318, -15943230, 885735, -19683, 243, 1, 434685788658, -178988265918, 25569752274, -1291401630, 23914845, -177147, 729, 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=3, q=3, r=1.

LINKS

Table of n, a(n) for n=0..35.

FORMULA

T(n,k) = A117253(n-k)*3^((n-k)*k). T(n,k) = [prod_{j=0..n-k-1}(1-3*j)]/(n-k)!*3^(n*(n-1)/2-k*(k-1)/2) for n>k>=0, with T(n,n) = 1.

EXAMPLE

Triangle T begins:

1;

1,1;

-3,3,1;

45,-27,9,1;

-2430,1215,-243,27,1;

433026,-196830,32805,-2187,81,1;

-245525742,105225318,-15943230,885735,-19683,243,1;

434685788658,-178988265918,25569752274,-1291401630,23914845,-177147,729,1;

Matrix cube T^3 has powers of 3 in the 2nd diagonal:

1;

3,1;

0,9,1;

0,0,27,1;

0,0,0,81,1;

0,0,0,0,243,1;

0,0,0,0,0,729,1; ...

PROG

(PARI) {T(n, k)=local(m=1, p=3, q=3, r=1); prod(j=0, n-k-1, m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

CROSSREFS

Cf. A117253 (column 0); variants: A117250 (p=q=2), 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: A065431 A271082 A053375 * A332498 A181407 A114187

Adjacent sequences:  A117249 A117250 A117251 * A117253 A117254 A117255

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Mar 14 2006

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified June 5 16:30 EDT 2020. Contains 334852 sequences. (Running on oeis4.)