A356718 T(n,k) is the total number of prime factors, counted with multiplicity, of k!*(n-k)!, for 0 <= k <= n. Triangle read by rows.

0, 0, 0, 1, 0, 1, 2, 1, 1, 2, 4, 2, 2, 2, 4, 5, 4, 3, 3, 4, 5, 7, 5, 5, 4, 5, 5, 7, 8, 7, 6, 6, 6, 6, 7, 8, 11, 8, 8, 7, 8, 7, 8, 8, 11, 13, 11, 9, 9, 9, 9, 9, 9, 11, 13, 15, 13, 12, 10, 11, 10, 11, 10, 12, 13, 15, 16, 15, 14, 13, 12, 12, 12

Offset: 0

Author

Dario T. de Castro, Aug 24 2022

Keywords

• nonn
• tabl
• easy
• look

Comments

k!*(n-k)! is the denominator in binomial(n,k) = n!/(k!*(n-k)!) and all prime factors in the denominator cancel to leave an integer, so that T(n,k) = A022559(n) - A132896(n,k).

Examples

			Triangle begins:
  n\k| 0  1  2  3  4  5  6  7
  ---+--------------------------------------
   0 | 0
   1 | 0, 0;
   2 | 1, 0, 1;
   3 | 2, 1, 1, 2;
   4 | 4, 2, 2, 2, 4;
   5 | 5, 4, 3, 3, 4, 5;
		

Links

• Dario T. de Castro, Rows n = 0..140 of triangle, flattened

Crossrefs

Cf. A007318, A001222, A022559, A132896, A303279.

Programs

Mathematica

T[n_,k_]:=PrimeOmega[Factorial[k]*Factorial[n-k]];
tab=Flatten[Table[T[n,k],{n,0,10},{k,0,n}]]

Formula

T(n,k) = bigomega(k!*(n-k)!), where 0 <= k <= n.

T(n,0) = T(n,n) = A022559(n).