0,12

Triangle T(n,k), 0<=k<=n, read by rows given by [1,-1,1,0,0,0,0,0,0,...] DELTA [1,0,0,0,0,0,...] where DELTA is the operator defined in A084938 . Riordan array (1/(1-x-x^2),(x*(1-x))/(1-x-x^2)) ; inverse array is (1/(1+x),(x/(1+x))*c(x/(1+x))) where c(x)is g.f. of A000108 .

The sequence is the set of matrix Markov coefficients for the determinant equals trace 2x2 matrix m = {{a, 1}, {-1, 1}}, which gives is an entirely different and new method of obtaining this sequence. Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 15 2009

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 15 2009: (Start)

Row sums are ( with the addition of a first row {0}):

{0, 1, 2, 2, 0, -4, -8, -8, 0, 16, 32,...}

This sequence is closely related to A167925. (End)

Sum{k, 0<=k<=n}T(n,k)*x^k = (-1)^n*A057093(n),(-1)^n*A057092(n), (-1)^n*A057091(n), (-1)^n*A057090(n), (-1)^n*A057089(n), (-1)^n*A057088(n), (-1)^n*A057087(n), (-1)^n*A030195(n+1), (-1)^n*A002605(n), A039834(n+1), A000007(n), A010892(n), A099087(n), A057083(n), A001787(n+1), A030191(n), A030192(n), A030240(n), A057084(n), A057085(n), A057086(n) for x=-11, -10, ..., 8, 9 respectively . Sum{k, 0<=k<=n}T(n,k)*A000045(k)=A100334(n) . Sum{k, 0<=k<=[n/2]}T(n-k,k)= A050935(n+2) . T(n,k)= Sum{j, j>=0}A109466(n,j)*binomial(j,k) .

Triangle begins:

1;

1, 1;

0, 1, 1;

-1, -1, 1, 1;

-1, -3, -2, 1, 1;

0, -2, -5, -3, 1, 1;

1, 2, -2, -7, -4, 1, 1;

1, 5, 7, -1, -9, -5, 1, 1;

0, 3, 12, 15, 1, -11, -6, 1, 1;

-1, -3, 3, 21, 26, 4, -13, -7, 1, 1;

-1, -7, -15, -3, 31, 40, 8, -15, -8, 1, 1;

Contribution from Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 15 2009: (Start)

Clear[m, a, n, v];

m = {{a, 1}, {-1, 1}};

v[0] := {0, 1};

v[n_] := v[n] = m.v[n - 1];

Table[CoefficientList[v[n][[1]], a], {n, 0, 10}];

Flatten[%] (End)

Cf. A063967.

Sequence in context: A109865 A096874 A090046 this_sequence A035103 A155033 A107889

Cf. A167925 [From Roger L. Bagula (rlbagulatftn(AT)yahoo.com), Nov 15 2009]

Adjacent sequences: A129264 A129265 A129266 this_sequence A129268 A129269 A129270

sign,tabl,new

Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Jun 08 2007