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

1,7

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

Links

Harvey P. Dale, Table of n, a(n) for n = 1..1000

L. Shapiro, A survey of the Riordan group

Formula

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

Example

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)

Mathematica

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

Crossrefs

Cf. A007318, A128308.

Sequence in context: A343730 A343761 A336423 * A349394 A034369 A368096

Adjacent sequences: A128304 A128305 A128306 * A128308 A128309 A128310

Keyword

nonn,tabl

Author

Gary W. Adamson, Feb 25 2007

Extensions

More terms from Harvey P. Dale, Nov 25 2018

Status

approved