A104530 Expansion of (1+sqrt(1-4x))/(4sqrt(1-4x)-2).

1, 3, 15, 78, 411, 2178, 11574, 61596, 328083, 1748346, 9319650, 49688004, 264943902, 1412826132, 7534329804, 40180417848, 214285867011, 1142822599722, 6094924220394, 32505784870068, 173362418710506, 924593002335612, 4931138212856244, 26299312319336328, 140262655976846766

Offset: 0

Author

Paul Barry, Mar 12 2005

Keywords

• easy
• nonn

Comments

Apply the Riordan matrix ((1+sqrt(1-4x))/2,(1-sqrt(1-4x))/2) (inverse of (1/(1-x),x(1-x))) to 4^n.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200

Crossrefs

Cf. A088218, A067336, A076025.

Programs

Mathematica

CoefficientList[Series[(1+Sqrt[1-4*x])/(4*Sqrt[1-4*x]-2), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 17 2012 *)

PARI

x='x+O('x^66); Vec((1+sqrt(1-4*x))/(4*sqrt(1-4*x)-2)) \\ Joerg Arndt, May 13 2013

Formula

a(n) = 0^n + Sum_{k=0..n} 3^(k+1)*C(2*n-1, n-k-1)*2*(k+1)/(n+k+1).

Recurrence: 3*n*a(n) = 2*(14*n-9)*a(n-1) - 32*(2*n-3)*a(n-2). - Vaclav Kotesovec, Oct 17 2012

a(n) ~ 2^(4*n-1)/3^n. - Vaclav Kotesovec, Oct 17 2012