A048621 -id:A048621 - cp's OEIS frontend

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

0, 0, 1, 1, 2, 3, 4, 3, 4, 4, 5, 5, 6, 7, 9, 7, 8, 8, 10, 9, 10, 10, 11, 11, 11, 12, 12, 11, 13, 13, 14, 11, 13, 14, 14, 13, 14, 14, 16, 15, 16, 18, 19, 19, 19, 19, 21, 19, 20, 19, 21, 20, 21, 21, 21, 19, 20, 20, 22, 22, 24, 25, 25, 23, 23, 23, 24, 24, 27, 26, 27, 25, 27, 28, 29, 28

Offset: 0

Labos Elemer, Mar 28 2003

It is easy to show that a(n) is between n/log(n) and 2n/log(n) (for n>n0), cf. [Campbell 1984]. The sequence A137687, roughly the middle of this interval, is a fair approximation for A081399. See A137686 for the (signed) difference of the two sequences.

M. F. Hasler, Table of n, a(n) for n = 0..3000.
Douglas M. Campbell, The Computation of Catalan Numbers, Mathematics Magazine, Vol. 57, No. 4. (Sep., 1984), pp. 195-208.

Cf. A001222, A000108, A022559, A023816, A048621, A081405, A120626, A137686-A137687.

Cf. A080405.

with(numtheory):a:=proc(n) if n=0 then 0 else bigomega(binomial(2*n,n)/(1+n)) fi end: seq(a(n), n=0..75); # Zerinvary Lajos, Apr 11 2008

a[n_] := PrimeOmega[ CatalanNumber[n]]; Table[a[n], {n, 0, 75}] (* Jean-François Alcover, Jul 02 2013 *)

A081399(n)=bigomega(prod(i=2,n,(n+i)/i)) \\ M. F. Hasler, Feb 06 2008

a(n)=A001222[A000108(n)]

Edited and extended by M. F. Hasler, Feb 06 2008

A048622 Difference of maximal and central values of A001222 when applied to {C(n,k)} set.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 2, 1, 2, 1, 0, 0, 2, 1, 0, 0, 0, 0, 0, 1, 3, 2, 1, 1, 3, 2, 1, 0, 2, 1, 2, 1, 1, 0, 0, 0, 2, 2, 1, 0, 1, 1, 3, 2, 3, 2, 0, 0, 2, 0, 0, 0, 4, 3, 4, 3, 2, 2, 3, 3, 5, 4, 3, 2, 2, 1, 2, 1, 3, 2, 1, 1, 2, 1, 0, 0, 1, 1, 3, 2, 1, 0, 0, 0, 3, 2, 2, 2, 4, 2, 2, 2, 3, 2

Offset: 1

Labos Elemer

nonn

			n=24: the sums of prime factor exponents when k runs from 0 to 24 are {0,4,4,5,5,7,6,8,6,8,8,9,7,9,8,8,6,8,6,7,5,5,4,4,0}. The central value is 7, the maximal is 9 so a(24)=9-7.

Michel Marcus, Table of n, a(n) for n = 1..5000

Cf. A001222, A046660, A020740, A048620, A048621.

a(n) = vecmax(apply(bigomega, vector(n+1, k, binomial(n,k-1)))) - bigomega(binomial(n, n\2)); \\ Michel Marcus, Jun 25 2021

a(n) = Max_k {A001222(C(n, k))} - A001222(A001405(n)).

a(n) = A048620(n) - A048621(n). - Sean A. Irvine, Jun 24 2021

cp's OEIS Frontend

A081399 Bigomega of the n-th Catalan number: a(n) = A001222(A000108(n)).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions