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A112555 Triangle T, read by rows, such that the m-th matrix power satisfies T^m = I + m*(T - I) and consequently the matrix logarithm satisfies LOG(T) = T - I, where I is the identity matrix. +0
7
1, 1, 1, -1, 0, 1, 1, 1, 1, 1, -1, -2, -2, 0, 1, 1, 3, 4, 2, 1, 1, -1, -4, -7, -6, -3, 0, 1, 1, 5, 11, 13, 9, 3, 1, 1, -1, -6, -16, -24, -22, -12, -4, 0, 1, 1, 7, 22, 40, 46, 34, 16, 4, 1, 1, -1, -8, -29, -62, -86, -80, -50, -20, -5, 0, 1, 1, 9, 37, 91, 148, 166, 130, 70, 25, 5, 1, 1, -1, -10, -46, -128, -239, -314, -296, -200, -95, -30, -6, 0 (list; table; graph; listen)
OFFSET

0,12

COMMENT

Signed version of A108561. Row sums equal A084247. The n-th unsigned row sum = A001045(n) + 1 (Jacobsthal numbers). Central terms of even-indexed rows are a signed version of A072547. Sums of squared terms in rows yields A112556, which equals the first differences of the unsigned central terms.

Equals row reversal of triangle A112468 up to sign, where A112468 is the Riordan array (1/(1-x),x/(1+x)). - Paul D. Hanna (pauldhanna(AT)juno.com), Jan 20 2006

FORMULA

G.f.: 1/(1-x*y) + x/((1-x*y)*(1+x+x*y)); the m-th matrix power T^m has the g.f.: 1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)). Recurrence: T(n, k) = [T^-1](n-1, k) + [T^-1](n-1, k-1), where T^-1 is the matrix inverse of T.

EXAMPLE

Triangle T begins:

1;

1,1;

-1,0,1;

1,1,1,1;

-1,-2,-2,0,1;

1,3,4,2,1,1;

-1,-4,-7,-6,-3,0,1;

1,5,11,13,9,3,1,1;

-1,-6,-16,-24,-22,-12,-4,0,1;

1,7,22,40,46,34,16,4,1,1;

-1,-8,-29,-62,-86,-80,-50,-20,-5,0,1; ...

Matrix log, LOG(T) = T - I, begins:

0;

1,0;

-1,0,0;

1,1,1,0;

-1,-2,-2,0,0;

1,3,4,2,1,0;

-1,-4,-7,-6,-3,0,0; ...

Matrix inverse, T^-1 = 2*I - T, begins:

1;

-1,1;

1,0,1;

-1,-1,-1,1;

1,2,2,0,1;

-1,-3,-4,-2,-1,1; ...

where adjacent sums in row n of T^-1 gives row n+1 of T.

PROGRAM

(PARI) {T(n, k)=local(x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff((1+2*x+x*y)/((1-x*y)*(1+x+x*y)), n, X), k, Y)}

(PARI) {T(n, k)=local(m=1, x=X+X*O(X^n), y=Y+Y*O(Y^k)); polcoeff(polcoeff(1/(1-x*y) + m*x/((1-x*y)*(1+x+x*y)), n, X), k, Y)} (Hanna)

CROSSREFS

Cf. A108561, A084247, A001045, A072547, A112556.

Cf. A112468 (reversed rows).

Adjacent sequences: A112552 A112553 A112554 this_sequence A112556 A112557 A112558

Sequence in context: A137581 A113414 A112185 this_sequence A108561 A104579 A079531

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna (pauldhanna(AT)juno.com), Sep 21 2005

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Last modified October 11 13:47 EDT 2008. Contains 144830 sequences.


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