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

0,1

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

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.

Example

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

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 *)

Prog

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

Crossrefs

Cf. A047999, A002487.

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

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

Sequence in context: A321016 A077051 A115955 * A106346 A296212 A189921

Adjacent sequences: A106341 A106342 A106343 * A106345 A106346 A106347

Keyword

easy,nonn,tabl

Author

Paul Barry, Apr 29 2005

Status

approved