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Triangle read by rows: T(n,k) is phi(binomial(n,k)), where phi is Euler's totient function (0 <= k <= n).
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%I #15 Sep 08 2022 08:45:16

%S 1,1,1,1,1,1,1,2,2,1,1,2,2,2,1,1,4,4,4,4,1,1,2,8,8,8,2,1,1,6,12,24,24,

%T 12,6,1,1,4,12,24,24,24,12,4,1,1,6,12,24,36,36,24,12,6,1,1,4,24,32,48,

%U 72,48,32,24,4,1,1,10,40,80,80,120,120,80,80,40,10,1,1,4,20,80,240,240

%N Triangle read by rows: T(n,k) is phi(binomial(n,k)), where phi is Euler's totient function (0 <= k <= n).

%C Row n contains n+1 terms. Row sums yield A064450.

%F T(n, k) = A000010(A007318(n, k)) (0 <= k <= n).

%F T(2n,n) = A066973(n).

%e T(6,3)=8 because the positive integers relatively prime to binomial(6,3)=20 and not exceeding 20 are 1,3,7,9,11,13,17 and 19.

%e Triangle begins:

%e 1;

%e 1, 1;

%e 1, 1, 1;

%e 1, 2, 2, 1;

%e 1, 2, 2, 2, 1;

%e 1, 4, 4, 4, 4, 1;

%p with(numtheory): T:=(n,k)->phi(binomial(n,k)): for n from 0 to 12 do seq(T(n,k),k=0..n) od; # yields sequence in triangular form

%t Flatten[Table[EulerPhi[Binomial[n, k]], {n, 0, 12}, {k, 0, n}]] (* _Vincenzo Librandi_, May 01 2019 *)

%o (Magma) /* As triangle */ [[EulerPhi(Binomial(n,k)): k in [0..n]]: n in [0.. 10]]; // _Vincenzo Librandi_, May 01 2019

%Y Cf. A000010 (totient), A007318 (binomial).

%Y Cf. A064450, A066973.

%K nonn,tabl

%O 0,8

%A _Emeric Deutsch_, Feb 06 2005