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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.

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A361039 a(n) = 55440 * (3*n)!/((2*n)!*(n+4)!).

Original entry on oeis.org

2310, 1386, 2310, 5544, 16335, 55055, 204204, 813960, 3432198, 15142050, 69334650, 327523680, 1588667850, 7883530578, 39904290580, 205532444040, 1075067283906, 5701114384350, 30608320603770, 166169731127400, 911270544740325
Offset: 0

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Author

Peter Bala, Mar 04 2023

Keywords

Comments

Compare with the super ballot numbers A348893(n) = 840*(2*n)!/(n!*(n+4)!).

Crossrefs

Cf. A000139, A001764, A007226, A007272, A361037, A361038, A361041.

Programs

  • Maple
    seq(  55440 * (3*n)!/((2*n)!*(n+4)!), n = 0..20);

Formula

a(n) = 2310*binomial(3*n,n) - 2057*binomial(3*n,n+1) + 627*binomial(3*n,n+2) - 102*binomial(3*n,n+3) + 7*binomial(3*n, n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 4)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 2310.
a(n) ~ (27/4)^n * 27720*sqrt(3/Pi)/n^(9/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 9)*A'(x) + (6*x + 8)*A(x) - 18480 = 0, with A(0) = 2310 and A'(0) = 1386.

A361040 a(n) = 420*(3*n)!/(n!*(2*n + 3)!).

Original entry on oeis.org

70, 21, 30, 70, 210, 735, 2856, 11970, 53130, 246675, 1187550, 5890248, 29954680, 155602020, 823184880, 4424618730, 24116031162, 133072694475, 742405558650, 4182821562150, 23776769743650, 136248095712855, 786482994679200
Offset: 0

Views

Author

Peter Bala, Mar 04 2023

Keywords

Comments

The Catalan numbers A000108 are defined by the formula Catalan(n) = (2*n)!/(n!*(n+1)!). Gessel (1992) considered generalized Catalan numbers defined by Catalan(r,n) = J(r)*(2*n)!/(n!*(n+r+1)!), where J(r) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. Gessel's generalized Catalan numbers are particular cases of super ballot numbers. See A135573 for a table of these generalized Catalan numbers.
For r = 0,1,2,..., it appears that there is an integer C(r) such the sequence {C(r)*(3*n)!/(n!*(2*n + r)!) : n >= 0} is integral. This is the case r = 3. For other cases see A005809 (r = 0, C(0) = 1), A001764 (r = 1, C(1) = 1), A000139 (r = 2, C(2) = 4) and A361041 (r = 4, C(4) = 1680).

Links

Crossrefs

Cf. A000139, A001764, A005809, A007272, A135573, A361038, A361041.

Programs

  • Maple
    seq( 420*(3*n)!/(n!*(2*n + 3)!), n = 0..20)

Formula

a(n) = 70*binomial(3*n,2*n) - 189*binomial(3*n,2*n+1) + 114*binomial(3*n,2*n+2) -32*binomial(3*n,2*n+3). Thus a(n) is an integer.
P-recursive: 2*(n + 1)*(2*n + 3)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 70.
a(n) ~ (27/4)^n * 105*sqrt(3/(16*Pi))/n^(7/2).
The o.g.f. A(x) satisfies the differential equation
x^2*(4 - 27*x^4)*A''(x) + 2*x*(7 - 27*x)*A'(x) + (6 - 6*x)*A(x) - 420 = 0, with A(0) = 70 and A'(0) = 21.
Showing 1-2 of 2 results.

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