This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A059268 #54 Feb 25 2025 05:12:11
%S A059268 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,
%T A059268 8,16,32,64,128,1,2,4,8,16,32,64,128,256,1,2,4,8,16,32,64,128,256,512,
%U A059268 1,2,4,8,16,32,64,128,256,512,1024,1,2,4,8,16,32,64,128,256,512,1024,2048
%N A059268 Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...
%C A059268 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).
%C A059268 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
%C A059268 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
%H A059268 Reinhard Zumkeller, <a href="/A059268/b059268.txt">Rows n = 0..150 of triangle, flattened</a>
%H A059268 J. L. Arregui, <a href="https://arxiv.org/abs/math/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%H A059268 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A059268 E.g.f.: exp(x+2*y) (T coordinates).
%F A059268 a(n) = A018900(n+1) - A140513(n). - _Reinhard Zumkeller_, Jun 24 2009
%F A059268 T(n,k) = A173786(n-1,k-1) - A173787(n-1,k-1), 0<k<=n. - _Reinhard Zumkeller_, Feb 28 2010
%F A059268 T(n,k) = 2^k. - _Reinhard Zumkeller_, Jan 29 2010
%F A059268 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
%F A059268 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
%e A059268 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}.
%e A059268 Triangle starts:
%e A059268 1;
%e A059268 1, 2;
%e A059268 1, 2, 4;
%e A059268 1, 2, 4, 8;
%e A059268 1, 2, 4, 8, 16;
%e A059268 1, 2, 4, 8, 16, 32;
%e A059268 ...
%p A059268 seq(seq(2^k,k=0..n),n=0..10);
%t A059268 Table[2^k, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2013 *)
%o A059268 (Haskell)
%o A059268 a059268 n k = a059268_tabl !! n !! k
%o A059268 a059268_row n = a059268_tabl !! n
%o A059268 a059268_tabl = iterate (scanl (+) 1) [1]
%o A059268 -- _Reinhard Zumkeller_, Apr 18 2013, Jul 05 2012
%o A059268 (Python)
%o A059268 from math import isqrt
%o A059268 def A059268(n):
%o A059268 a = (m:=isqrt(k:=n+1<<1))-(k<=m*(m+1))
%o A059268 return 1<<n-((a+1)*a>>1) # _Chai Wah Wu_, Feb 24 2025
%Y A059268 Cf. A140531.
%Y A059268 Cf. A000079.
%Y A059268 Cf. A131816.
%Y A059268 Row sums give A126646.
%K A059268 nonn,tabl,easy
%O A059268 0,3
%A A059268 _N. J. A. Sloane_, Jan 23 2001
%E A059268 Formula corrected by _Reinhard Zumkeller_, Feb 23 2010