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 A130321 #32 Jan 27 2025 02:30:28
%S A130321 1,2,1,4,2,1,8,4,2,1,16,8,4,2,1,32,16,8,4,2,1,64,32,16,8,4,2,1,128,64,
%T A130321 32,16,8,4,2,1,256,128,64,32,16,8,4,2,1,512,256,128,64,32,16,8,4,2,1,
%U A130321 1024,512,256,128,64,32,16,8,4,2,1,2048,1024,512,256,128,64,32,16,8,4,2,1
%N A130321 Triangle, (2^0, 2^1, 2^2, ...) in every column.
%C A130321 A130321^2 = A130322. Binomial transform of A130321 = triangle A027649. A007318^2 = A038207 = A007318(n,k) * A130321(n,k); i.e., the square of Pascal's triangle = dot product of Pascal's triangle rows and A130321 rows: A007318^2 = (1; 2,1; 4,4,1; 8,12,6,1;...), where row 3, (8,12,6,1) = (1,3,3,1) dot (8,4,2,1).
%C A130321 Sequence B is called a reverse reluctant sequence of sequence A, if B is triangle array read by rows: row number k lists first k elements of the sequence A in reverse order. Sequence A130321 is the reverse reluctant sequence of sequence of power of 2 (A000079). - _Boris Putievskiy_, Dec 13 2012
%C A130321 From _Wolfdieter Lang_, Jan 10 2015: (Start)
%C A130321 This is the Riordan array (1/(1-2*x), x).
%C A130321 Row sums give A000225(n+1) = 2^(n+1) - 1.
%C A130321 Alternating row sums give A001045(n+1).
%C A130321 The inverse Riordan array is (1-2*x, x) = A251635. (End)
%H A130321 Reinhard Zumkeller, <a href="/A130321/b130321.txt">Rows n = 0..120 of triangle, flattened</a>
%H A130321 Boris Putievskiy, <a href="http://arxiv.org/abs/1212.2732">Transformations Integer Sequences And Pairing Functions</a> arXiv:1212.2732 [math.CO], 2012.
%F A130321 Triangle, (1, 2, 4, 8, ...) in every column. Rows are reversals of A059268 terms.
%F A130321 a(n)=2^m, where m=(t*t + 3*t + 4)/2 - n, t=floor((-1+sqrt(8*n-7))/2). - _Boris Putievskiy_, Dec 13 2012
%F A130321 From _Wolfdieter Lang_, Jan 10 2015: (Start)
%F A130321 T(n, m) = 2^(n-m) if n >= m >= 0 and 0 otherwise.
%F A130321 G.f. of row polynomials R(n,x) = Sum_{m=0..n} 2^(n-m)*x^m is 1/((1-2*z)*(1-x*z)) (Riordan property).
%F A130321 G.f. column m (with leading zeros) x^m/(1-2*x), m >= 0.
%F A130321 The diagonal sequences are D(k) = repeat(2^k), k >= 0. (End)
%e A130321 The triangle T(n,m) begins:
%e A130321 n\m 0 1 2 3 4 5 6 7 8 9 10 ...
%e A130321 0: 1
%e A130321 1: 2 1
%e A130321 2: 4 2 1
%e A130321 3: 8 4 2 1
%e A130321 4: 16 8 4 2 1
%e A130321 5: 32 16 8 4 2 1
%e A130321 6: 64 32 16 8 4 2 1
%e A130321 7: 128 64 32 16 8 4 2 1
%e A130321 8: 256 128 64 32 16 8 4 2 1
%e A130321 9: 512 256 128 64 32 16 8 4 2 1
%e A130321 10: 1024 512 256 128 64 32 16 8 4 2 1
%e A130321 ... Reformatted. - _Wolfdieter Lang_, Jan 10 2015
%t A130321 T[n_, m_] := 2^(n-m);
%t A130321 Table[T[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* _Jean-François Alcover_, Aug 07 2018 *)
%o A130321 (Haskell)
%o A130321 a130321 n k = a130321_tabl !! n !! k
%o A130321 a130321_row n = a130321_tabl !! n
%o A130321 a130321_tabl = iterate (\row -> (2 * head row) : row) [1]
%o A130321 -- _Reinhard Zumkeller_, Feb 27 2013
%Y A130321 Cf. A059268, A027649, A130322, A038207, A131816, A000225, A001045, A251635.
%K A130321 nonn,tabl,easy
%O A130321 0,2
%A A130321 _Gary W. Adamson_, May 24 2007
%E A130321 More terms from _Philippe Deléham_, Feb 08 2009