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

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

Author

Peter Bala, Mar 04 2023

Keywords

• nonn
• easy

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.