A059268 Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...

1, 1, 2, 1, 2, 4, 1, 2, 4, 8, 1, 2, 4, 8, 16, 1, 2, 4, 8, 16, 32, 1, 2, 4, 8, 16, 32, 64, 1, 2, 4, 8, 16, 32, 64, 128, 1, 2, 4, 8, 16, 32, 64, 128, 256, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024, 2048

Offset: 0

Author

N. J. A. Sloane, Jan 23 2001

Keywords

• nonn
• tabl
• easy

Comments

Triangular array T(n,k) read by rows, where T(n,k) = i!*j! times coefficient of x^n*y^k in exp(x+2y).

T(n,k) is the number of subsets of {0,1,...,n} whose largest element is k. To see this, let A be any subset of the 2^k subsets of {0,1,...,k-1}. Then there are 2^k subsets of the form (A U {k}). See example below. - Dennis P. Walsh, Nov 27 2011

Sequence B is called a reluctant sequence of sequence A, if B is triangle array read by rows: row number k coincides with first k elements. A059268 is reluctant sequence of sequence A000079. - Boris Putievskiy, Dec 17 2012

Examples

			T(4,3)=8 since there are 8 subsets of {0,1,2,3,4} whose largest element is 3, namely, {3}, {0,3}, {1,3}, {2,3}, {0,1,3}, {0,2,3}, {1,2,3}, and {0,1,2,3}.
Triangle starts:
  1;
  1, 2;
  1, 2, 4;
  1, 2, 4, 8;
  1, 2, 4, 8, 16;
  1, 2, 4, 8, 16, 32;
  ...
		

Links

• Reinhard Zumkeller, Rows n = 0..150 of triangle, flattened
• J. L. Arregui, Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles, arXiv:math/0109108 [math.NT], 2001.
• Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Crossrefs

Cf. A140531.

Cf. A000079.

Cf. A131816.

Row sums give A126646.

Programs

Haskell

a059268 n k = a059268_tabl !! n !! k
a059268_row n = a059268_tabl !! n
a059268_tabl = iterate (scanl (+) 1) [1]
-- Reinhard Zumkeller, Apr 18 2013, Jul 05 2012

Maple

seq(seq(2^k,k=0..n),n=0..10);

Mathematica

Table[2^k, {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jun 10 2013 *)

Python

from math import isqrt
def A059268(n):
    a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
    return 1<>1) # Chai Wah Wu, Feb 24 2025

Formula

E.g.f.: exp(x+2*y) (T coordinates).

a(n) = A018900(n+1) - A140513(n). - Reinhard Zumkeller, Jun 24 2009

T(n,k) = A173786(n-1,k-1) - A173787(n-1,k-1), 0Reinhard Zumkeller, Feb 28 2010

T(n,k) = 2^k. - Reinhard Zumkeller, Jan 29 2010

As a linear array, the sequence is a(n) = 2^(n-1-t*(t+1)/2), where t = floor((-1+sqrt(8*n-7))/2), n>=1. - Boris Putievskiy, Dec 17 2012

As a linear array, the sequence is a(n) = 2^(n-1-t*(t+1)/2), where t = floor(sqrt(2*n)-1/2), n>=1. - Zhining Yang, Jun 09 2017

Extensions

Formula corrected by Reinhard Zumkeller, Feb 23 2010