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T(n,k) = number of subsets of {1,..., n} containing exactly k primes, triangle read by rows, 0<=k<n.
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%I #8 Nov 04 2020 09:38:07

%S 2,2,2,2,4,2,4,8,4,0,4,12,12,4,0,8,24,24,8,0,0,8,32,48,32,8,0,0,16,64,

%T 96,64,16,0,0,0,32,128,192,128,32,0,0,0,0,64,256,384,256,64,0,0,0,0,0,

%U 64,320,640,640,320,64,0,0,0,0,0,128,640,1280,1280,640,128,0,0,0,0,0

%N T(n,k) = number of subsets of {1,..., n} containing exactly k primes, triangle read by rows, 0<=k<n.

%C T(n,k) = T(n, A000720(n)-k) for 0<=k<=A000720(n);

%C T(n,k) = 0 iff k > A000720(n);

%C A089819(n) = T(n,0); A089821(n) = T(n,1) for n>1; A089822(n) = T(n,2) for n>2;

%C A089820(n) = Sum(T(n,k): 1<=k<=A000720(n));

%C T(n,k) = A007318(A000720(n),k) * A000079(n-A000720(n)).

%F T(n, k) = binomial(pi(n), k)*2^(n-pi(n)), with pi = A000720.

%t T[n_, k_] := Binomial[PrimePi[n], k] 2^(n - PrimePi[n]);

%t Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* _Jean-François Alcover_, Nov 04 2020 *)

%Y Cf. A000040.

%K nonn,tabl

%O 1,1

%A _Reinhard Zumkeller_, Nov 12 2003