A130321 Triangle, (2^0, 2^1, 2^2, ...) in every column.

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

Offset

0,2

Comments

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).

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

From Wolfdieter Lang, Jan 10 2015: (Start)

This is the Riordan array (1/(1-2*x), x).

Row sums give A000225(n+1) = 2^(n+1) - 1.

Alternating row sums give A001045(n+1).

The inverse Riordan array is (1-2*x, x) = A251635. (End)

Links

Reinhard Zumkeller, Rows n = 0..120 of triangle, flattened

Boris Putievskiy, Transformations Integer Sequences And Pairing Functions arXiv:1212.2732 [math.CO], 2012.

Formula

Triangle, (1, 2, 4, 8, ...) in every column. Rows are reversals of A059268 terms.

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

From Wolfdieter Lang, Jan 10 2015: (Start)

T(n, m) = 2^(n-m) if n >= m >= 0 and 0 otherwise.

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).

G.f. column m (with leading zeros) x^m/(1-2*x), m >= 0.

The diagonal sequences are D(k) = repeat(2^k), k >= 0. (End)

Example

The triangle T(n,m) begins:
  n\m     0   1   2   3  4  5  6  7  8  9 10 ...
  0:      1
  1:      2   1
  2:      4   2   1
  3:      8   4   2   1
  4:     16   8   4   2  1
  5:     32  16   8   4  2  1
  6:     64  32  16   8  4  2  1
  7:    128  64  32  16  8  4  2  1
  8:    256 128  64  32 16  8  4  2  1
  9:    512 256 128  64 32 16  8  4  2  1
 10:   1024 512 256 128 64 32 16  8  4  2  1
 ... Reformatted. - Wolfdieter Lang, Jan 10 2015

Mathematica

T[n_, m_] := 2^(n-m);
Table[T[n, m], {n, 0, 11}, {m, 0, n}] // Flatten (* Jean-François Alcover, Aug 07 2018 *)

Prog

(Haskell)
a130321 n k = a130321_tabl !! n !! k
a130321_row n = a130321_tabl !! n
a130321_tabl = iterate (\row -> (2 * head row) : row) [1]
-- Reinhard Zumkeller, Feb 27 2013

Crossrefs

Cf. A059268, A027649, A130322, A038207, A131816, A000225, A001045, A251635.

Sequence in context: A138895 A138846 A235670 * A101508 A106471 A180870

Adjacent sequences: A130318 A130319 A130320 * A130322 A130323 A130324

Keyword

nonn,tabl,easy,changed

Author

Gary W. Adamson, May 24 2007

Extensions

More terms from Philippe Deléham, Feb 08 2009

Status

approved