A128307 Triangle, (1, 0, 1, 2, 4, 8, ...) in every column.

1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 4, 2, 1, 0, 1, 8, 4, 2, 1, 0, 1, 16, 8, 4, 2, 1, 0, 1, 32, 16, 8, 4, 2, 1, 0, 1, 64, 32, 16, 8, 4, 2, 1, 0, 1, 128, 64, 32, 16, 8, 4, 2, 1, 0, 1, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0, 1, 512, 256, 128, 64, 32, 16, 8, 4, 2, 1, 0

Offset: 1

Author

Gary W. Adamson, Feb 25 2007

Keywords

• nonn
• tabl

Comments

Row sums = (1, 1, 2, 4, 8, ...). A128308 = binomial transform of A128307.

Riordan array ( 1 + x^2/(1 - 2*x), x ). T(n,k) gives the number of compositions of n of the form 1 + 1 + ... + 1 + a_1 + ... + a_m beginning with k 1's and with a_1 > 1. See Shapiro, Section 5. An example is given below. - Peter Bala, Aug 18 2014

Examples

			First few rows of the triangle:
  1;
  0, 1;
  1, 0, 1;
  2, 1, 0, 1;
  4, 2, 1, 0, 1;
  8, 4, 2, 1, 0, 1;
  ...
From _Peter Bala_, Aug 18 2014: (Start)
Row 4: [4,2,1,0,1]
              Compositions                Number
k = 0     4, 3 + 1, 2 + 2, 2 + 1 + 1        4
k = 1     1 + 3, 1 + 2 + 1                  2
k = 2     1 + 1 + 2                         1
k = 3                                       0
k = 4     1 + 1 + 1 + 1                     1
(End)
		

Links

• Harvey P. Dale, Table of n, a(n) for n = 1..1000
• L. Shapiro, A survey of the Riordan group

Crossrefs

Cf. A007318, A128308.

Programs

Mathematica

Join[{1,0,1},Table[Join[NestWhileList[#/2&,2^n,#!=1&],{0,1}],{n,0,10}]]//Flatten (* Harvey P. Dale, Nov 25 2018 *)

Formula

(1, 0, 1, 2, 4, 8, ...) in every column.

Extensions

More terms from Harvey P. Dale, Nov 25 2018