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

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

Offset: 0

Author

Peter Bala, Mar 01 2023

Keywords

• nonn
• easy

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).