A153342 Binomial transform of triangle A046854 (shifted).

1, 2, 0, 4, 1, 0, 8, 4, 1, 0, 16, 12, 5, 1, 0, 32, 32, 18, 6, 1, 0, 64, 80, 56, 25, 7, 1, 0, 128, 192, 160, 88, 33, 8, 1, 0, 256, 448, 432, 280, 129, 42, 9, 1, 0, 512, 1024, 1120, 832, 450, 180, 52, 10, 1, 0, 1024, 2304, 2816, 2352, 1452, 681, 242, 63, 11, 1, 0

Offset: 0

Author

Gary W. Adamson, Dec 24 2008

Keywords

• nonn
• tabl

Comments

Row sums = odd indexed Fibonacci numbers.

Mirror image of triangle in A121462. - Philippe Deléham, Dec 31 2008

Triangle T(n,k), 0 <= k <= n, read by rows given by [2,0,0,0,0,0,0,0,0,0,0,0,...] DELTA [0,1/2,1/2,0,0,0,0,0,0,0,0,0,...] where DELTA is the operator defined in A084938. - Philippe Deléham, Jan 01 2009

Examples

			First few rows of the triangle =
    1;
    2,    0;
    4,    1,    0;
    8,    4,    1,   0;
   16,   12,    5,   1,   0;
   32,   32,   18,   6,   1,   0;
   64,   80,   56,  25,   7,   1,  0;
  128,  192,  160,  88,  33,   8,  1,  0;
  256,  448,  432, 280, 129,  42,  9,  1, 0;
  512, 1024, 1120, 832, 450, 180, 52, 10, 1, 0;
  ...
		

Links

• Rigoberto Flórez, Leandro Junes, Luisa M. Montoya, and José L. Ramírez, Counting Subwords in Non-Decreasing Dyck Paths, J. Int. Seq. (2025) Vol. 28, Art. No. 25.1.6. See pp. 8, 19.

Crossrefs

Cf. A046854, A001519.

Formula

Triangle read by rows, A007318 * A046854 (shifted down 1 row, inserting a "1" at (0,0).

G.f.: (1-y*x)/(1-2*x-y*x+y*x^2). - Philippe Deléham, Mar 27 2012

T(n,k) = 2*T(n-1,l) + T(n-1,k-1) - T(n-2,k-1), T(0,0) = 1, T(1,0) = 2, T(1,1) = 0 and T(n,k) = 0 if k<0 or if k>n. - Philippe Deléham, Mar 27 2012

Extensions

Second term corrected by Philippe Deléham, Jan 01 2009