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%I #74 Dec 14 2021 22:57:39
%S 1,1,0,1,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,1,1,1,1,1,1,0,1,0,1,0,1,0,
%T 1,0,1,1,0,1,1,0,1,1,0,1,0,1,0,0,0,1,0,1,0,1,1,1,1,1,1,1,1,1,1,0,1,0,
%U 0,0,1,0,1,0,0,0,1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,1,0
%N Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).
%C Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). - _Gary W. Adamson_, May 20 2007
%C Characteristic function of A169581: a(A169581(n)) = 1; a(A169582(n)) = 0. - _Reinhard Zumkeller_, Dec 02 2009
%C The function T(n,k) = T(k,n) is defined for k > n but only the values for 1 <= k <= n as a triangular array are listed here.
%C T(n,k) = |K(n-k|k)| where K(i|j) is the Kronecker symbol. - _Peter Luschny_, Aug 05 2012
%C Twice the sum over the antidiagonals, starting with entry T(n-1,1), for n >= 3, is the same as the row n sum (i.e., phi(n): 2*Sum_{k=1..floor(n/2)} T(n-k,k) = phi(n), n >= 3). - _Wolfdieter Lang_, Apr 26 2013
%C The number of zeros in the n-th row of the triangle is cototient(n) = A051953(n). - _Omar E. Pol_, Apr 21 2017
%C This triangle is the j = 1 sub-triangle of A349221(n,k) = Sum_{j>=1} [k|binomial(n-1,k-1) AND gcd(n,k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. - _Richard L. Ollerton_, Dec 14 2021
%H Reinhard Zumkeller, <a href="/A054521/b054521.txt">Rows n = 1..125 of triangle, flattened</a>
%H Jakub Jaroslaw Ciaston, <a href="/plot2a?name1=A054531&name2=A164306&tform1=untransformed&tform2=untransformed&shift=0&radiop1=xy&drawpoints=true">A054531 vs A164306</a> (plot shows these ones)
%H <a href="/index/Ch#char_fns">Index entries for characteristic functions</a>
%F T(n,k) = A063524(A050873(n,k)). - _Reinhard Zumkeller_, Dec 02 2009, corrected Sep 03 2015
%F T(n,k) = A054431(n,k) = A054431(k,n). - _R. J. Mathar_, Jul 21 2016
%e The triangle T(n,k) begins:
%e n\k 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 ...
%e 1: 1
%e 2: 1 0
%e 3: 1 1 0
%e 4: 1 0 1 0
%e 5: 1 1 1 1 0
%e 6: 1 0 0 0 1 0
%e 7: 1 1 1 1 1 1 0
%e 8: 1 0 1 0 1 0 1 0
%e 9: 1 1 0 1 1 0 1 1 0
%e 10: 1 0 1 0 0 0 1 0 1 0
%e 11: 1 1 1 1 1 1 1 1 1 1 0
%e 12: 1 0 0 0 1 0 1 0 0 0 1 0
%e 13: 1 1 1 1 1 1 1 1 1 1 1 1 0
%e 14: 1 0 1 0 1 0 0 0 1 0 1 0 1 0
%e 15: 1 1 0 1 0 0 1 1 0 0 1 0 1 1 0
%e ... (Reformatted by _Wolfdieter Lang_, Apr 26 2013)
%e Sums over antidiagonals: n = 3: 2*T(2,1) = 2 = T(3,1) + T(3,2) = phi(3). n = 4: 2*(T(3,1) + T(2,2)) = 2 = phi(4), etc. - _Wolfdieter Lang_, Apr 26 2013
%p A054521_row := n -> seq(abs(numtheory[jacobi](n-k,k)),k=1..n);
%p for n from 1 to 13 do A054521_row(n) od; # _Peter Luschny_, Aug 05 2012
%t T[ n_, k_] := Boole[ n>0 && k>0 && GCD[ n, k] == 1] (* _Michael Somos_, Jul 17 2011 *)
%t T[ n_, k_] := If[ n<1 || k<1, 0, If[ k>n, T[ k, n], If[ k==1, 1, If[ n>k, T[ k, Mod[ n, k, 1]], 0]]] (* _Michael Somos_, Jul 17 2011 *)
%o (PARI) {T(n, k) = n>0 && k>0 && gcd(n, k)==1} /* _Michael Somos_, Jul 17 2011 */
%o (Sage)
%o def A054521_row(n): return [abs(kronecker_symbol(n-k,k)) for k in (1..n)]
%o for n in (1..13): print(A054521_row(n)) # _Peter Luschny_, Aug 05 2012
%o (Haskell)
%o a054521 n k = a054521_tabl !! (n-1) !! (k-1)
%o a054521_row n = a054521_tabl !! (n-1)
%o a054521_tabl = map (map a063524) a050873_tabl
%o a054521_list = concat a054521_tabl
%o -- _Reinhard Zumkeller_, Sep 03 2015
%Y Cf. A051731, A054522, A215200.
%Y Cf. A050873, A063524.
%Y Cf. A349221.
%K nonn,tabl
%O 1,1
%A _N. J. A. Sloane_, Apr 09 2000
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