A037031 Number of combinations of n objects taken pi(n) at a time.

1, 1, 2, 3, 6, 10, 20, 35, 70, 126, 210, 462, 792, 1716, 3003, 5005, 8008, 19448, 31824, 75582, 125970, 203490, 319770, 817190, 1307504, 2042975, 3124550, 4686825, 6906900, 20030010, 30045015, 84672315, 129024480, 193536720, 286097760, 417225900, 600805296

Offset: 0

Author

G. L. Honaker, Jr.

Keywords

• nonn

Comments

Number of subsets of [n] in which exactly half of the elements are primes: a(6) = 20: {}, {1,2}, {1,3}, {1,5}, {2,4}, {2,6}, {3,4}, {3,6}, {4,5}, {5,6}, {1,2,3,4}, {1,2,3,6}, {1,2,4,5}, {1,2,5,6}, {1,3,4,5}, {1,3,5,6}, {2,3,4,6}, {2,4,5,6}, {3,4,5,6}, {1,2,3,4,5,6}. - Alois P. Heinz, Oct 11 2022

Examples

			C(16,6) = 8008.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2000 (terms n = 1..1000 from Vincenzo Librandi)

Crossrefs

Cf. A000040, A000720.

Programs

Maple

with(numtheory); A037031:=n->binomial(n, pi(n)); seq(A037031(k), k=0..100); # Wesley Ivan Hurt, Oct 19 2013

Mathematica

Table[Binomial[n,PrimePi[n]],{n,40}] (* Harvey P. Dale, Mar 22 2012 *)

Python

from math import comb
from sympy import primepi
def A037031(n): return comb(n,primepi(n)) # Chai Wah Wu, Oct 14 2022

Formula

a(n) = C(n,pi(n)).

Extensions

a(0)=1 prepended by Alois P. Heinz, Oct 11 2022