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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. 46
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; refs; listen; history; text; internal format)
OFFSET

0,12

COMMENTS

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, Jan 20 2006

The elements here match A108561 in absolute value, but the signs are crucial to the properties that the matrix A112555 exhibits; the main property being T^m = I + m*(T - I). This property is not satisfied by A108561. - Paul D. Hanna, Nov 10 2009

Eigensequence of the triangle = A140165 [Gary W. Adamson, Jan 30 2009]

Triangle T(n,k), read by rows, given by [1,-2,0,0,0,0,0,0,0,...] DELTA [1,0,-1,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. [Philippe Deléham, Sep 17 2009]

LINKS

Paul D. Hanna, Table of n, a(n) for n = 0..1080

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.

Sum_{k, 0<=k<=n} T(n,k)*x^(n-k)= A165760(n), A165759(n), A165758(n), A165755(n), A165752(n), A165746(n), A165751(n), A165747(n), A000007(n), A000012(n), A084247(n), A165553(n), A165622(n), A165625(n), A165638(n), A165639(n), A165748(n), A165749(n), A165750(n) for x= -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. [Philippe Deléham, Oct 07 2009]

Sum_{k, 0<=k<=n} T(n,k)*x^k = A166157(n), A166153(n), A166152(n), A166149(n), A166036(n), A166035(n), A091004(n+1), A077925(n), A000007(n), A165326(n), A084247(n), A165405(n), A165458(n), A165470(n), A165491(n), A165505(n), A165506(n), A165510(n), A165511(n) for x = -9,-8,-7,-6,-5,-4,-3,-2,-1,0,1,2,3,4,5,6,7,8,9 respectively. [Philippe Deléham, Oct 08 2009]

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.

MATHEMATICA

Clear[t]; t[0, 0] = 1; t[n_, 0] = (-1)^(Mod[n, 2]+1); t[n_, n_] = 1; t[n_, k_] /; k == n-1 := t[n, k] = Mod[n, 2]; t[n_, k_] /; 0 < k < n-1 := t[n, k] = -t[n-1, k] - t[n-1, k-1]; Table[t[n, k], {n, 0, 13}, {k, 0, n}] // Flatten (* Jean-François Alcover, Mar 06 2013 *)

PROG

(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)}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

(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)}

for(n=0, 12, for(k=0, n, print1(T(n, k), ", ")); print(""))

(Sage)

def A112555_row(n):

    @cached_function

    def prec(n, k):

        if k==n: return 1

        if k==0: return 0

        return -prec(n-1, k-1)-sum(prec(n, k+i-1) for i in (2..n-k+1))

    return [(-1)^(n-k+1)*prec(n+1, k) for k in (1..n+1)]

for n in (0..12): print A112555_row(n) # Peter Luschny, Mar 16 2016

CROSSREFS

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

Cf. A112468 (reversed rows).

Cf. A140165 [From Gary W. Adamson, Jan 30 2009]

Sequence in context: A192062 A172371 A279006 * A108561 A174626 A264909

Adjacent sequences:  A112552 A112553 A112554 * A112556 A112557 A112558

KEYWORD

sign,tabl

AUTHOR

Paul D. Hanna, Sep 21 2005

STATUS

approved

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Last modified May 28 17:57 EDT 2017. Contains 287241 sequences.