A305612 Expansion of 1/2 * (((1 + 2*x)/(1 - 2*x))^(3/2) - 1).

0, 3, 9, 22, 51, 114, 250, 540, 1155, 2450, 5166, 10836, 22638, 47124, 97812, 202488, 418275, 862290, 1774630, 3646500, 7482618, 15334748, 31391724, 64194312, 131151566, 267711444, 546031500, 1112864200, 2266587900, 4613409000, 9384609960, 19079454960

Offset: 0

Author

Seiichi Manyama, Jun 06 2018

Keywords

• nonn

Comments

Let 1/2 * (((1 + k*x)/(1 - k*x))^(m/k) - 1) = a(0) + a(1)*x + a(2)*x^2 + ...

Then n*a(n) = 2*m*a(n-1) + k^2*(n-2)*a(n-2) for n > 1.

Links

• Seiichi Manyama, Table of n, a(n) for n = 0..3000

Crossrefs

1/2 * (((1 + 2*x)/(1 - 2*x))^(m/2) - 1): A001405(n-1) (m=1), this sequence (m=3).

Cf. A305031.

Programs

Maple

seq(coeff(series((1/2)*(((1+2*x)/(1-2*x))^(3/2)-1), x,n+1),x,n),n=0..35); # Muniru A Asiru, Jun 06 2018

Mathematica

CoefficientList[Series[((((1+2x)/(1-2x))^(3/2))-1)/2,{x,0,40}],x] (* Harvey P. Dale, Nov 04 2020 *)

Formula

n*a(n) = 6*a(n-1) + 4*(n-2)*a(n-2) for n > 1.

a(n) = A305031(n)/2 for n > 0.