A004332 -id:A004332 - cp's OEIS frontend

A023828 Sum of exponents in prime-power factorization of C(4n,n-2).

0, 3, 5, 5, 5, 9, 9, 11, 8, 8, 10, 12, 13, 15, 15, 16, 13, 16, 17, 16, 16, 18, 20, 22, 18, 20, 21, 20, 20, 22, 24, 25, 24, 25, 26, 26, 24, 28, 28, 29, 24, 28, 29, 29, 28, 29, 32, 34, 31, 31, 33, 31, 31, 34, 32, 34, 30, 33, 34, 37, 37, 40, 43, 42, 37, 39, 40, 41, 41, 41, 43, 44, 40, 45, 46, 46, 45, 47

Offset: 2

Clark Kimberling

nonn

Ivan Neretin, Table of n, a(n) for n = 2..10000

Cf. A001222, A004332, A023826, A023827, A023829, A023830, A023831, A023832, A023833, A023834, A023835, A023836.

Join[{0}, Table[Total[FactorInteger[Binomial[4 n, n - 2]][[All, 2]]], {n, 3, 78}]] (* Ivan Neretin, Nov 02 2017 *)
a[n_] := PrimeOmega[Binomial[4*n, n-2]]; Array[a, 100, 2] (* Amiram Eldar, Jun 13 2025 *)

a(n) = bigomega(binomial(4*n,n-2)); \\ Amiram Eldar, Jun 13 2025

From Amiram Eldar, Jun 13 2025 (Start)
a(n) = A001222(A004332(n)).
a(n) = A023827(n) - A001222(3*n+2) + A001222(n-1). (End)

Offset corrected to 2 by Ivan Neretin, Nov 02 2017

A173621 Triangle of Generalized Runyon numbers R_{n,k}^(4) read by rows.

Original entry on oeis.org

1, 1, 4, 1, 12, 22, 1, 24, 120, 140, 1, 40, 380, 1140, 969, 1, 60, 920, 5060, 10626, 7084, 1, 84, 1890, 16380, 61425, 98280, 53820, 1, 112, 3472, 43400, 251720, 704816, 906192, 420732, 1, 144, 5880, 99960, 824670, 3518592, 7791168, 8347680, 3362260

Offset: 1

R. J. Mathar, Nov 08 2010

The Runyon numbers R_{n,k}^(1) are A001263, R_{n,k}^(2) are A108767. Row sums are in A002294.

			The triangle starts in row n=1 as
1;
1, 4;
1, 12, 22;
1, 24, 120, 140;
1, 40, 380, 1140, 969;
1, 60, 920, 5060, 10626, 7084;
1, 84, 1890, 16380, 61425, 98280, 53820;
1, 112, 3472, 43400, 251720, 704816, 906192, 420732;

Chunwei Song, The Generalized Schroeder Theory, El. J. Combin. 12 (2005) #R53

T(n,k) = R(n,k,4) with R(n,k,m)= binomial(n,k)*binomial(m*n,k-1)/n, 1<=k<=n.
T(n,n) = A002293(n).
T(n,n-1) = A004332(n).
T(n,2) = A046092(n-1).

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A023828 Sum of exponents in prime-power factorization of C(4n,n-2).

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A173621 Triangle of Generalized Runyon numbers R_{n,k}^(4) read by rows.

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