cp's OEIS Frontend

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.

Showing 1-4 of 4 results.

A361031 a(n) = (3^3)*(1*2*4*5*7*8*10*11)*(3*n)!/(n!*(n+4)!^2).

Original entry on oeis.org

11550, 2772, 4620, 15840, 81675, 550550, 4492488, 42325920, 446185740, 5148297000, 63985977000, 846321189120, 11802213457650, 172255143129300, 2615726247519000, 41127042052404000, 666874986879730860, 11114583114662181000, 189866473537245687000, 3316382259894423720000
Offset: 0

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Author

Peter Bala, Mar 01 2023

Keywords

Comments

Row 3 of A361027.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 840*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3)*(n + 4) and the result 840*(2*n)!/(n!*(n+4)!) is the super ballot number A348893(n). Similarly, the de Bruijn numbers A006480(n) = (3*n)!/n!^3 have the property that 6652800 * A006480(n) is divisible by ((n + 1)*(n + 2)*(n + 3)*(n + 4))^2.

Crossrefs

Row 3 of A361027. Cf. A006480, A348893, A361028, A361029, A361030.

Programs

  • Maple
    a := proc(n) option remember; if n = 0 then 11550 else
3*(3*n-1)*(3*n-2)/(n+4)^2*a(n-1) end if; end proc:
seq(a(n), n = 0..20);

Formula

a(n) = (3^3)*(1*2*4*5*7*8*10*11)/((n+1)*(n+2)*(n+3)*(n+4))^2 * (3*n)!/n!^3.
a(n) = (1/3)*(1*2*4*5*7*8*10*11) * A006480(n+4)/((3*n + 1)*(3*n + 2)*(3*n + 4)* (3*n + 5)*(3*n + 7)*(3*n + 8)*(3*n + 10)*(3*n + 11)), where A006480(n) = (3*n)!/n!^3.
a(n) = (1/3)*27^(n+4)*binomial(10/3, n+4)*binomial(11/3, n+4).
a(n) = (1/7)*A348893(n)*A361039(n). It can be shown from this that a(n) is always an integer.
a(n) ~ sqrt(3)*3326400*(27^n)/(Pi*n^9).
P-recursive: (n + 4)^2*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 11550.
The o.g.f. A(x) satisfies the differential equation x^2*(1 - 27*x)*A''(x) + x*(9 - 54*x)*A'(x) + (16 - 6*x)*A(x) - 184800 = 0, with A(0) = 11550 and A'(0) = 2772.

A361038 a(n) = 1680 * (3*n)!/((2*n)!*(n+3)!).

Original entry on oeis.org

280, 210, 420, 1176, 3960, 15015, 61880, 271320, 1248072, 5965050, 29414700, 148874400, 770263200, 4061212722, 21765976680, 118336861720, 651555929640, 3627981880950, 20405547069180, 115815267149400, 662742214356600
Offset: 0

Views

Author

Peter Bala, Mar 04 2023

Keywords

Comments

Compare with the super ballot numbers A007272(n) = 60*(2*n)!/(n!*(n+3)!).

Crossrefs

Cf. A000139, A001764, A007226, A007272, A361037, A361039, A361040.

Programs

  • Maple
    seq( 1680 * (3*n)!/((2*n)!*(n+3)!), n = 0..20);

Formula

a(n) = 280*binomial(3*n,n) - 228*binomial(3*n,n+1) + 54*binomial(3*n,n+2) - 5*binomial(3*n,n+3). Thus a(n) is an integer.
P-recursive: 2*(n + 3)*(2*n - 1) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 280.
a(n) ~ (27/4)^n * 840*sqrt(3/Pi)/n^(7/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 7)*A'(x) + (6*x + 6)*A(x) - 1680 = 0, with A(0) = 280 and A'(0) = 210.

A361037 a(n) = 20*(3*n)!/((2*n)!*(n+2)!).

Original entry on oeis.org

10, 10, 25, 84, 330, 1430, 6630, 32300, 163438, 852150, 4552275, 24812400, 137547000, 773564328, 4405019090, 25357898940, 147375745990, 863805209750, 5101386767295, 30332569967700, 181465130121450, 1091677288630950
Offset: 0

Views

Author

Peter Bala, Mar 04 2023

Keywords

Comments

Gessel (1992) introduced sequences {b(r,n): n >= 0} of super ballot numbers defined by b(r,n) = J(r) * (2*n)!/(n!*(n + r + 1)!), r = 0,1,2,..., where J(r) = (2*r + 2)!/(2*(r + 1)!) = (2^r)*Product_{j = 0..r} (2*j + 1) is chosen so that these numbers are always integers. The sequence {b(1,n) : n >= 0} is A000108, the sequence of Catalan numbers. See A135573 for a table of these generalized Catalan numbers.
We carry out an analogous construction using the numbers B(n) = A005809(n) = binomial(3*n,n) = (3*n)!/((2*n)!*n!) in place of the central binomial numbers. We define B(r,n), r = 0,1,2, ..., by B(r,n) = F(r) * (3*n)!/((2*n)!*(n + r + 1)!), where F(r) is the minimal choice to produce integer values for these quantities for all n. This sequence is the case r = 1. See A007226 (r = 0), A361038 (r = 2) and A361039 (r = 3).

Links

Crossrefs

Cf. A000139, A001764, A005809, A007226, A361038, A361039.

Programs

  • Maple
    seq( 20*(3*n)!/((2*n)!*(n+2)!), n = 0..20);
  • Mathematica
    Table[20 (3n)!/((2n)!(n+2)!),{n,0,30}] (* Harvey P. Dale, Aug 05 2024 *)

Formula

a(n) = 10*binomial(3*n,n) - 7*binomial(3*n,n+1) + binomial(3*n,n+2). Thus a(n) is an integer.
P-recursive: 2*(n + 2)(2*n - 1)*a(n) = 3*(3*n - 1)*(3*n - 2)*a(n-1) with a(0) = 10.
a(n) ~ (27/4)^n * 10*sqrt(3/Pi)/n^(5/2).
The o.g.f. satisfies the differential equation
x^2*(27*x - 4)*A''(x) + 2*x*(27*x - 5)*A'(x) + 2*(3*x + 2)*A(x) - 40 = 0, with A(0) = 10 and A'(0) = 10.

A361041 a(n) = 1680*(3*n)!/(n!*(2*n + 4)!).

Original entry on oeis.org

70, 14, 15, 28, 70, 210, 714, 2660, 10626, 44850, 197925, 906192, 4279240, 20746936, 102898110, 520543380, 2679559018, 14007652050, 74240555865, 398363958300, 2161524522150, 11847660496770, 65540249556600, 365634339159024
Offset: 0

Views

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, A005809, A348893, A361039, A361040.

Programs

  • Maple
    seq( 1680*(3*n)!/(n!*(2*n + 4)!), n = 0..20);

Formula

a(n) = 70*binomial(3*n,2*n) - 196*binomial(3*n,2*n+1) + 141*binomial(3*n,2*n+2) - 65*binomial(3*n,2*n+3) + 14*binomial(3*n,2*n+4). Thus a(n) is an integer.
P-recursive: 2*(n + 2)*(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/(4*Pi))/n^(9/2).
The o.g.f. A(x) satisfies the differential equation
x^2*(4 - 27*x^4)*A''(x) + 2*x*(9 - 27*x)*A'(x) + (12 - 6*x)*A(x) - 840 = 0, with A(0) = 70 and A'(0) = 14.
Showing 1-4 of 4 results.

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