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A117254 Triangle T, read by rows, where matrix power T^4 has powers of 4 in the secondary diagonal: [T^4](n+1,n) = 4^(n+1), with all 1's in the main diagonal and zeros elsewhere. 8
1, 1, 1, -6, 4, 1, 224, -96, 16, 1, -39424, 14336, -1536, 64, 1, 30277632, -10092544, 917504, -24576, 256, 1, -98180268032, 31004295168, -2583691264, 58720256, -393216, 1024, 1, 1321338098679808, -402146377859072, 31748398252032, -661424963584, 3758096384, -6291456, 4096, 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=4, q=4, r=1.

LINKS

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

FORMULA

T(n,k) = A117255(n-k)*4^((n-k)*k). T(n,k) = (-1)^(n-k)*4^(n*(n-1)/2-k*(k-1)/2)/(n-k)!*prod_{j=0..n-k-1}(4*j-1) for n>k>=0, with T(n,n) = 1.

EXAMPLE

Triangle T begins:

1;

1,1;

-6,4,1;

224,-96,16,1;

-39424,14336,-1536,64,1;

30277632,-10092544,917504,-24576,256,1;

-98180268032,31004295168,-2583691264,58720256,-393216,1024,1; ...

Matrix power T^4 has powers of 4 in the 2nd diagonal:

1;

4,1;

0,16,1;

0,0,64,1;

0,0,0,256,1;

0,0,0,0,1024,1;

0,0,0,0,0,4096,1; ...

PROG

(PARI) {T(n, k)=local(m=1, p=4, q=4, r=1); prod(j=0, n-k-1, m*r-p*j)/(n-k)!*q^((n-k)*(n+k-1)/2)}

CROSSREFS

Cf. A117255 (column 0); variants: A117250 (p=q=2), A117252 (p=q=3), 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: A166905 A278071 A132870 * A211022 A021613 A157701

Adjacent sequences:  A117251 A117252 A117253 * A117255 A117256 A117257

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Mar 14 2006

STATUS

approved

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Last modified January 16 23:43 EST 2022. Contains 350377 sequences. (Running on oeis4.)