A089818 T(n,k) = number of subsets of {1,..., n} containing exactly k primes, triangle read by rows, 0<=k

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, 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, 64, 320, 640, 640, 320, 64, 0, 0, 0, 0, 0, 128, 640, 1280, 1280, 640, 128, 0, 0, 0, 0, 0

Offset: 1

Author

Reinhard Zumkeller, Nov 12 2003

Keywords

• nonn
• tabl

Comments

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

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

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

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

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

Crossrefs

Cf. A000040.

Programs

Mathematica

T[n_, k_] := Binomial[PrimePi[n], k] 2^(n - PrimePi[n]);
Table[T[n, k], {n, 1, 12}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 04 2020 *)

Formula

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