A023992 -id:A023992 - cp's OEIS frontend

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).

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, 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, 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

Offset: 0

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 0..10000

Cf. A001222, A000897, A006480, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023991, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[2*n, n, n]] - PrimeOmega[Multinomial[n, n, n]]; Array[a, 100, 0] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((4*n)!/((2*n)!*n!^2)) - bigomega((3*n)!/(n!^3)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A023980(n) - A023978(n) = A001222(A000897(n)) - A001222(A006480(n)).
a(n) = A022559(4*n) + 2*A022559(n) - A022559(2*n) - A022559(3*n). (End)

Name clarified, offset changed to 0 and a(0) prepended by Amiram Eldar, Jun 11 2025

A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

Original entry on oeis.org

1, 4, 6, 8, 8, 12, 12, 13, 13, 15, 17, 20, 19, 22, 21, 22, 21, 24, 25, 26, 25, 31, 30, 32, 30, 31, 33, 33, 32, 36, 34, 36, 34, 36, 36, 37, 36, 40, 40, 42, 40, 45, 48, 49, 49, 51, 50, 52, 49, 50, 50, 53, 50, 56, 53, 53, 53, 55, 58, 60, 59, 62, 60, 60, 55, 58, 59, 61, 60, 65, 62, 65, 63, 66, 69, 68

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A022559, A023978, A076191, A248707.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023992, A023993.

a[n_] := PrimeOmega[Multinomial[n+1, n, n-1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / ((n-1)!*n!*(n+1)!)); \\ Amiram Eldar, Jun 11 2025

From Amiram Eldar, Jun 11 2025: (Start)
a(n) = A001222(A248707(n)).
a(n) = A022559(3*n) - A022559(n-1) - A022559(n) - A022559(n+1) = A022559(3*n) - 3*A022559(n) - A001222(n+1) + A001222(n) = A023978(n) - A076191(n). (End)

A023993 Sum of exponents of primes in multinomial coefficient M(3n; n+2,n-1,n-1).

Original entry on oeis.org

0, 3, 6, 8, 8, 11, 11, 14, 14, 14, 17, 21, 18, 20, 22, 23, 21, 24, 24, 27, 26, 29, 29, 34, 29, 30, 35, 33, 32, 34, 33, 39, 34, 34, 37, 39, 35, 38, 41, 43, 40, 45, 46, 50, 51, 48, 49, 54, 49, 50, 51, 52, 49, 56, 53, 55, 54, 53, 58, 62, 57, 58, 61, 63, 56, 58, 58, 61, 61, 63, 62, 68, 61, 65, 70

Offset: 1

Clark Kimberling

nonn

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A001222, A022559.

Cf. A023978, A023979, A023980, A023981, A023982, A023983, A023984, A023985, A023986, A023987, A023990, A023991, A023992.

a[n_] := PrimeOmega[Multinomial[n + 2, n - 1, n - 1]]; Array[a, 100] (* Amiram Eldar, Jun 11 2025 *)

a(n) = bigomega((3*n)! / ((n+2)!*(n-1)!^2)); \\ Amiram Eldar, Jun 11 2025

a(n) = A022559(3*n) - A022559(n+2) - 2*A022559(n-1). - Amiram Eldar, Jun 11 2025

cp's OEIS Frontend

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).

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A023991 Sum of exponents of primes in multinomial coefficient M(3n; n+1,n,n-1).

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A023993 Sum of exponents of primes in multinomial coefficient M(3n; n+2,n-1,n-1).

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Mathematica

PARI

Formula