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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.

Showing 1-3 of 3 results.

A361033 a(n) = 3*(4*n)!/(n!*(n+1)!^3).

Original entry on oeis.org

3, 9, 280, 17325, 1513512, 162954792, 20193091776, 2768662192725, 409716429837000, 64358256798795960, 10605621798062141760, 1817833036248401270280, 321997225483126007438400, 58649494641569379926280000, 10941649720331183519046796800, 2084191938036600263793119045925
Offset: 0

Views

Author

Peter Bala, Mar 01 2023

Keywords

Comments

Row 0 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that A000984(n) is divisible by n + 1 and the result (2*n)!/(n!*(n+1)!) is the n-th Catalan number A000108(n). Similarly, the numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 3*A008977(n) is divisible by (n + 1)^3, leading to the present sequence. Cf. A361028. Do these numbers have a combinatorial interpretation?
Conjecture: a(n) is odd iff n = 2^k - 1 for some k >= 0.

Crossrefs

Cf. A000108, A007226, A007228, A008977, A361028, A361032, A361034, A361035.

Programs

  • Maple
    seq(3*(4*n)!/(n!*(n+1)!^3), n = 0..20);
  • Mathematica
    Table[3 (4n)!/(n! ((n+1)!)^3),{n,0,15}] (* Harvey P. Dale, Jul 30 2024 *)

Formula

a(n) = 3*A008977(n)/(n+1)^3.
a(n) = (3/4)*A008977(n+1)/((4*n+1)*(4*n+2)*(4*n+3)).
a(n) = (1/2)*A007228(n)*A007226(n)*A000108(n).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+1)^3 * a(n-1) with a(0) = 3.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(6 - 1152*x)*A(x)'' + x*(7 - 816*x)*A(x)' + (1 - 24*x)*A(x) - 3 = 0 with A(0) = 3, A'(0) = 9 and A''(0) = 560.
a(n) ~ 3*sqrt(1/(2*Pi^3)) * 2^(8*n)/n^(9/2).

A361034 a(n) = 2520*(4*n)!/(n!*(n+2)!^3).

Original entry on oeis.org

315, 280, 3675, 116424, 5885880, 399072960, 33129291195, 3190228041000, 344161801063080, 40616781150254400, 5155510596280207800, 695029472211496161600, 98570579229528369624000, 14597207555235045670540800, 2243893009052293495117018875, 356344642367340570239409729000
Offset: 0

Views

Author

Peter Bala, Mar 01 2023

Keywords

Comments

Row 1 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 6*A000984(n) is divisible by (n + 1)*(n + 2) and the result (2*n)!/(n!*(n+2)!) is the super ballot number A007054(n). Similarly, the numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 2520*A008977(n) is divisible by ((n + 1)*(n + 2))^3, leading to the present sequence. Cf. A361029.
Conjecture: a(n) is odd iff n = 2^k - 2 for some k >= 1.

Crossrefs

Cf. A007054, A008977, A361029, A361032, A361033, A361035.

Programs

  • Maple
    seq(2520*(4*n)!/(n!*(n+2)!^3), n = 0..20);

Formula

a(n) = 2520*A008977(n)/((n+1)*(n+2))^3.
a(n) = (315/2)*A008977(n+2)/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+2)^3 * a(n-1) with a(0) = 315.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(9 - 1152*x)*A(x)'' + x*(19 - 816*x)*A(x)' + (8 - 24*x)*A(x) - 2520 = 0 with A(0) = 315, A'(0) = 280 and A''(0) = 7350.
a(n) ~ 630*sqrt(8/Pi^3) * 2^(8*n)/n^(15/2).

A361035 a(n) = 9979200 * (4*n)!/(n!*(n+3)!^3).

Original entry on oeis.org

46200, 17325, 116424, 2134440, 67953600, 3086579925, 179961581800, 12633303042360, 1023952465972800, 93080123469333000, 9292590788015304000, 1003030870975774344000, 115656146295979953692160, 14112534648127632044761125, 1808633485822731984665865000
Offset: 0

Views

Author

Peter Bala, Mar 01 2023

Keywords

Comments

Row 2 of A361032.
The central binomial numbers A000984(n) = (2*n)!/n!^2 have the property that 60*A000984(n) is divisible by (n + 1)*(n + 2)*(n + 3) and the result (2*n)!/(n!*(n+3)!) is the super ballot number A007272(n). Similarly, the numbers A008977(n) = (4*n)!/n!^4 appear to have the property that 9979200*A008977(n) is divisible by ((n + 1)*(n + 2)*(n + 3))^3, leading to the present sequence. Cf. A361030.
Conjecture: a(n) is odd iff n = 2^k - 3 for some k >= 2.

Crossrefs

Cf. A007272, A008977, A361030, A361032, A361033, A361034.

Programs

  • Maple
    seq( 9979200 * (4*n)!/(n!*(n+3)!^3 ), n = 0..20);

Formula

a(n) = 9979200 * A008977(n)/((n+1)*(n+2)*(n+3))^3.
a(n) = (15925)*A008977(n+3)/((4*n+1)*(4*n+2)*(4*n+3)*(4*n+5)*(4*n+6)*(4*n+7)*(4*n+9)*(4*n+10)*(4*n+11)).
P-recursive: a(n) = 4*(4*n-1)*(4*n-2)*(4*n-3)/(n+3)^3 * a(n-1) with a(0) = 46200.
The o.g.f. A(x) satisfies the differential equation
x^3*(1 - 256*x)*A(x)''' + x^2*(12 - 1152*x)*A(x)'' + x*(37 - 816*x)*A(x)' + (27 - 24*x)*A(x) - 1247400 = 0 with A(0) = 46200, A'(0) = 17325 and A''(0) = 232848.
a(n) ~ 2494800*sqrt(8/Pi^3) * 2^(8*n)/n^(21/2).
Showing 1-3 of 3 results.

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