0,12

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

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 Hanna, Nov 10 2009

Eigensequence of the triangle = A140165 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), 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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 17 2009]

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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 08 2009]

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.

(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)

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

Cf. A112468 (reversed rows).

Cf. A140165 [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jan 30 2009]

Sequence in context: A156311 A113414 A112185 this_sequence A108561 A104579 A079531

Adjacent sequences: A112552 A112553 A112554 this_sequence A112556 A112557 A112558

sign,tabl,new

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