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(2^(n-m)*x^m, m=0..n) 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
KEYWORD
nonn,tabl
AUTHOR
Gary W. Adamson, May 24 2007
EXTENSIONS
More terms from Philippe Deléham, Feb 08 2009
STATUS
approved