This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A356718 #89 Mar 04 2025 23:15:23
%S A356718 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,
%T A356718 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,
%U A356718 10,12,13,15,16,15,14,13,12,12,12
%N 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.
%C A356718 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).
%H A356718 Dario T. de Castro, <a href="/A356718/b356718.txt">Rows n = 0..140 of triangle, flattened</a>
%F A356718 T(n,k) = bigomega(k!*(n-k)!), where 0 <= k <= n.
%F A356718 T(n,0) = T(n,n) = A022559(n).
%e A356718 Triangle begins:
%e A356718 n\k| 0 1 2 3 4 5 6 7
%e A356718 ---+--------------------------------------
%e A356718 0 | 0
%e A356718 1 | 0, 0;
%e A356718 2 | 1, 0, 1;
%e A356718 3 | 2, 1, 1, 2;
%e A356718 4 | 4, 2, 2, 2, 4;
%e A356718 5 | 5, 4, 3, 3, 4, 5;
%t A356718 T[n_,k_]:=PrimeOmega[Factorial[k]*Factorial[n-k]];
%t A356718 tab=Flatten[Table[T[n,k],{n,0,10},{k,0,n}]]
%Y A356718 Cf. A007318, A001222, A022559, A132896, A303279.
%K A356718 nonn,tabl,easy,look
%O A356718 0,7
%A A356718 _Dario T. de Castro_, Aug 24 2022