A359489 Expansion of 1/sqrt(1 - 4*x/(1-x)^3).

1, 2, 12, 68, 396, 2358, 14262, 87252, 538440, 3345434, 20899816, 131154264, 826135794, 5220372274, 33077821314, 210087769632, 1337104370320, 8525602760550, 54449281992528, 348250972411252, 2230296171922008, 14300414859019290, 91791793780179790

Offset: 0

Author

Seiichi Manyama, Mar 24 2023

Keywords

• nonn

Links

• Winston de Greef, Table of n, a(n) for n = 0..1218

Crossrefs

Cf. A085362, A110170, A162478, A359758, A360132.

Programs

Mathematica

CoefficientList[Series[1/Sqrt[1-(4x)/(1-x)^3],{x,0,30}],x] (* Harvey P. Dale, Aug 09 2023 *)

PARI

my(N=30, x='x+O('x^N)); Vec(1/sqrt(1-4*x/(1-x)^3))

PARI

a(n) = sum(k=0, n, binomial(2*k,k) * binomial(n+2*k-1,n-k)) \\ Winston de Greef, Mar 24 2023

Formula

a(n) = Sum_{k=0..n} binomial(2*k,k) * binomial(n+2*k-1,n-k).

n*a(n) = (8*n-6)*a(n-1) - (10*n-24)*a(n-2) + 4*(n-3)*a(n-3) - (n-4)*a(n-4) for n > 3.

a(n) ~ sqrt(2*(2 + (35 + 3*sqrt(129))^(1/3))) * (40 + 7*(262 + 6*sqrt(129))^(1/3) + (262 + 6*sqrt(129))^(2/3))^n / ((43*(86 + 6*sqrt(129)))^(1/6) * sqrt(Pi*n) * 3^n * (262 + 6*sqrt(129))^(n/3)). - Vaclav Kotesovec, Mar 25 2023

a(0) = 1; a(n) = (2/n) * Sum_{k=0..n-1} (n+k) * binomial(n+1-k,2) * a(k). - Seiichi Manyama, Mar 28 2023