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Triangle read by rows: T(n,k) = binomial(k,n-k) mod 2.
15

%I #26 May 21 2022 08:31:24

%S 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,

%T 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,

%U 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

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

%C A skew version of Sierpinski’s triangle A047999. - _Johannes W. Meijer_, Jun 05 2011

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

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

%H G. C. Greubel, <a href="/A106344/b106344.txt">Rows n = 0..50, flattened</a>

%H Thomas Baruchel, <a href="https://doi.org/10.1007/s42979-019-0049-1">Flattening Karatsuba's Recursion Tree into a Single Summation</a>, SN Computer Science (2020) Vol. 1, Article No. 48.

%H George Beck and Karl Dilcher, <a href="https://arxiv.org/abs/2106.10400">A Matrix Related to Stern Polynomials and the Prouhet-Thue-Morse Sequence</a>, arXiv:2106.10400 [math.CO], 2021. See (1.6) p. 2.

%e Triangle begins

%e 1;

%e 0, 1;

%e 0, 1, 1;

%e 0, 0, 0, 1;

%e 0, 0, 1, 1, 1;

%e 0, 0, 0, 1, 0, 1;

%p seq(seq(`mod`(binomial(k, n-k), 2), k = 0..n), n = 0..15); # _G. C. Greubel_, Feb 07 2020

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

%o (PARI) T(n,k) = binomial(k,n-k)%2;

%o for(n=0,15, for(k=0,n, print1(T(n,k), ", "))) \\ _G. C. Greubel_, Feb 07 2020

%o (Magma) [ Binomial(k,n-k) mod 2: k in [0..n], n in [0..15]]; // _G. C. Greubel_, Feb 07 2020

%o (Sage) [[ mod(binomial(k,n-k), 2) for k in (0..n)] for n in (0..15)] # _G. C. Greubel_, Feb 07 2020

%o (GAP) Flat(List([0..15], n-> List([0..n], k-> (Binomial(k,n-k) mod 2) ))); # _G. C. Greubel_, Feb 07 2020

%Y Cf. A047999, A002487.

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

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

%K easy,nonn,tabl

%O 0,1

%A _Paul Barry_, Apr 29 2005