A154312 Triangle T(n,k), 0<=k<=n, read by rows, given by [0,1/2,-1/2,0,0,0,0,0,0,0,...] DELTA [2,-1/2,-1/2,2,0,0,0,0,0,0,0 ...] where DELTA is the operator defined in A084938 .

1, 0, 2, 0, 1, 3, 0, 0, 3, 5, 0, 0, 0, 7, 9, 0, 0, 0, 0, 15, 17, 0, 0, 0, 0, 0, 31, 33, 0, 0, 0, 0, 0, 0, 63, 65, 0, 0, 0, 0, 0, 0, 0, 127, 129, 0, 0, 0, 0, 0, 0, 0, 0, 255, 257, 0, 0, 0, 0, 0, 0, 0, 0, 0, 511, 513, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1023, 1025, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2047

Offset

0,3

Comments

Column sums give A003945.

Links

Table of n, a(n) for n=0..89.

Formula

Sum_{k, 0<=k<=n}T(n,k)*x^(n-k)= A040000(n), A094373(n), A000079(n), A083329(n), A095121(n), A154117(n), A131128(n), A154118(n), A131130(n), A154251(n), A154252(n) for x = -1,0,1,2,3,4,5,6,7,8,9 respectively.

G.f.: (1-x*y+x^2*y-x^2*y^2)/(1-3*x*y+2*x^2*y^2). - Philippe Deléham, Nov 02 2013

T(n,k) = 3*T(n-1,k-1) - 2*T(n-2,k-2), T(0,0) = 1, T(1,0) = 0, T(1,1) = 2, T(2,0) = 0, T(2,1) = 1, T(2,2) = 3, T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Nov 02 2013

Example

Triangle begins:
1;
0, 2;
0, 1, 3;
0, 0, 3, 5;
0, 0, 0, 7, 9;
0, 0, 0, 0, 15, 17; ...

Crossrefs

Cf. A000225, A094373

Sequence in context: A132213 A202502 A219839 * A236076 A364021 A363899

Adjacent sequences: A154309 A154310 A154311 * A154313 A154314 A154315

Keyword

nonn,tabl

Author

Philippe Deléham, Jan 07 2009

Status

approved