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

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

Offset

0,1

Comments

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

Links

Table of n, a(n) for 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.

Maple

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

Crossrefs

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

Sequence in context: A177844 A289303 A241803 * A245208 A256708 A207075

Adjacent sequences: A361035 A361036 A361037 * A361039 A361040 A361041

Keyword

nonn,easy

Author

Peter Bala, Mar 04 2023

Status

approved