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A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).

%I A023990 #11 Jun 11 2025 09:14:15
%S A023990 0,1,1,1,2,2,1,3,3,4,4,1,2,4,3,3,5,4,4,3,4,7,4,4,5,7,5,4,5,5,6,6,7,7,
%T A023990 9,7,9,11,8,9,9,9,8,6,7,7,6,6,8,9,8,8,9,10,7,8,9,10,11,7,8,11,11,10,
%U A023990 12,13,13,12,11,14,12,11,13,14,12,12,14,14,13,14,14,15,15,13,14,14,11,10,13,12,13,13,12,15
%N A023990 Sum of exponents of primes in multinomial coefficient M(4n; 2n,n,n) - sum of exponents of primes in M(3n; n,n,n).
%H A023990 Amiram Eldar, <a href="/A023990/b023990.txt">Table of n, a(n) for n = 0..10000</a>
%F A023990 From _Amiram Eldar_, Jun 11 2025: (Start)
%F A023990 a(n) = A023980(n) - A023978(n) = A001222(A000897(n)) - A001222(A006480(n)).
%F A023990 a(n) = A022559(4*n) + 2*A022559(n) - A022559(2*n) - A022559(3*n). (End)
%t A023990 a[n_] := PrimeOmega[Multinomial[2*n, n, n]] - PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* _Amiram Eldar_, Jun 11 2025 *)
%o A023990 (PARI) a(n) = bigomega((4*n)!/((2*n)!*n!^2)) - bigomega((3*n)!/(n!^3)); \\ _Amiram Eldar_, Jun 11 2025
%Y A023990 Cf. A001222, A000897, A006480, A022559.
%Y A023990 Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023991, A023992, A023993.
%K A023990 nonn
%O A023990 0,5
%A A023990 _Clark Kimberling_
%E A023990 Name clarified, offset changed to 0 and a(0) prepended by _Amiram Eldar_, Jun 11 2025