|
%I #44 Mar 04 2024 19:19:02
%S 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 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 1,2,4,8,16,32,64,128,256,512,1024,1,2,4,8,16,32,64,128,256,512,1024,2048
%N Concatenate subsequences [2^0, 2^1, ..., 2^n] for n = 0, 1, 2, ...
%C 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 a(n) = A018900(n+1) - A140513(n). - _Reinhard Zumkeller_, Jun 24 2009
%C T(n,k) = A173786(n-1,k-1) - A173787(n-1,k-1), 0<k<=n. - _Reinhard Zumkeller_, Feb 28 2010
%C 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 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 Reinhard Zumkeller, <a href="/A059268/b059268.txt">Rows n = 0..150 of triangle, flattened</a>
%H J. L. Arregui, <a href="http://arXiv.org/abs/math.NT/0109108">Tangent and Bernoulli numbers related to Motzkin and Catalan numbers by means of numerical triangles</a>, arXiv:math/0109108 [math.NT], 2001.
%H Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F E.g.f.: exp(x+2*y) (T coordinates).
%F T(n,k) = 2^k. - _Reinhard Zumkeller_, Jan 29 2010
%F 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 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 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 Triangle starts:
%e 1;
%e 1, 2;
%e 1, 2, 4;
%e 1, 2, 4, 8;
%e 1, 2, 4, 8, 16;
%e 1, 2, 4, 8, 16, 32;
%e ...
%p seq(seq(2^k,k=0..n),n=0..10);
%t Table[2^k, {n, 0, 11}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 10 2013 *)
%o (Haskell)
%o a059268 n k = a059268_tabl !! n !! k
%o a059268_row n = a059268_tabl !! n
%o a059268_tabl = iterate (scanl (+) 1) [1]
%o -- _Reinhard Zumkeller_, Apr 18 2013, Jul 05 2012
%Y Cf. A140531.
%Y Cf. A000079.
%Y Cf. A131816.
%Y Row sums give A126646.
%K nonn,tabl
%O 0,3
%A _N. J. A. Sloane_, Jan 23 2001
%E Formula corrected by _Reinhard Zumkeller_, Feb 23 2010
|
