A106344 Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.

1, 0, 1, 0, 1, 1, 0, 0, 0, 1, 0, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1

Offset: 0

Author

Paul Barry, Apr 29 2005

Keywords

• easy
• nonn
• tabl

Comments

A skew version of Sierpinski’s triangle A047999. - Johannes W. Meijer, Jun 05 2011

Row sums are A002487(n+1). Diagonal sums are A106345. Inverse is A106346.

Triangle formed by reading T triangle mod 2 with T := A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167. - Philippe Deléham, Dec 18 2008

Examples

			Triangle begins
  1;
  0, 1;
  0, 1, 1;
  0, 0, 0, 1;
  0, 0, 1, 1, 1;
  0, 0, 0, 1, 0, 1;
		

Links

• G. C. Greubel, Rows n = 0..50, flattened
• Thomas Baruchel, Flattening Karatsuba's Recursion Tree into a Single Summation, SN Computer Science (2020) Vol. 1, Article No. 48.
• George Beck and Karl Dilcher, A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

Crossrefs

Cf. A047999, A002487.

Cf. A106345 (diagonal sums), A106346 (inverse).

Cf. A026729, A062110, A084938, A099093, A106344, A109466, A110517, A112883, A130167.

Programs

GAP

Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # G. C. Greubel, Feb 07 2020

Magma

[ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // G. C. Greubel, Feb 07 2020

Maple

seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # G. C. Greubel, Feb 07 2020

Mathematica

Table[Mod[Binomial[k, n-k], 2], {n,0,15}, {k,0,n}]//Flatten (* G. C. Greubel, Apr 18 2017 *)

PARI

T(n,k) = binomial(k,n-k)%2;
for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ G. C. Greubel, Feb 07 2020

Sage

[[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # G. C. Greubel, Feb 07 2020