A184018 Expansion of c(x/(1-x-x^2)) / (1-x-x^2), c(x) the g.f. of A000108.

1, 2, 6, 19, 67, 254, 1017, 4236, 18168, 79680, 355635, 1609912, 7373401, 34102976, 159055728, 747211753, 3532452169, 16792693562, 80224098381, 384948157635, 1854469572120, 8965866981294, 43488834409737, 211569299607282

Offset: 0

Author

Paul Barry, Jan 08 2011

Keywords

• nonn
• easy

Comments

Hankel transform is (9,-5) Somos-4 sequence A184019.

The radius of convergence r of the g.f. A(x) satisfies: r = (1-r-r^2)/4 = limit a(n)/a(n+1) = (sqrt(29)-5)/2 = 0.19258240... with A(r) = 1/(2*r) = (sqrt(29)+5)/4 = 2.59629120... - Paul D. Hanna, Sep 06 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..1000

Programs

Maple

A000108 := proc(n) binomial(2*n,n)/(n+1) ; end proc:
A037027 := proc(n,m) add( binomial(m+k,m)*binomial(k,n-k-m),k=0..n-m) ; end proc:
A184018 := proc(n) add( A037027(n,k)*A000108(k),k=0..n) ; end proc:
seq(A184018(n),n=0..10) ; # R. J. Mathar, Jan 11 2011

Mathematica

CoefficientList[Series[(1 - x - x^2 - Sqrt[1 - 6 x + 3 x^2 + 6 x^3 + x^4])/(2 x (1 - x - x^2)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 04 2014 *)

PARI

{a(n)=polcoeff((1-sqrt(1-4*x/(1-x-x^2 +O(x^(n+2)))))/(2*x),n)} /* Paul D. Hanna, Sep 06 2011 */

Formula

G.f.: ( 1 - x - x^2 - sqrt((1 - x - x^2)*(1 - 5*x - x^2)) )/( 2*x*(1 - x - x^2) ).

G.f.: 1/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1 - x - x^2 - x/(1-x/(1-... (continued fraction).

a(n) = Sum_{k=0..n} (Sum_{j=0..n-k} binomial(k+j,k)*binomial(j,n-k-j))*A000108(k) = Sum_{k=0..n} A037027(n,k)*A000108(k).

G.f. satisfies A(x) = 1/(1-x-x^2) + x*A(x)^2. - Paul D. Hanna, Sep 06 2011

Conjecture: (n+1)*a(n) + 2*(1-3*n)*a(n-1) + 3*(n-1)*a(n-2) + 2*(3*n-5)*a(n-3) + (n-3)*a(n-4) = 0. - R. J. Mathar, Nov 15 2011

a(n) ~ (27+5*sqrt(29)) * sqrt(54*sqrt(29)-290) * (5+sqrt(29))^n / (sqrt(Pi) * n^(3/2) * 2^(n+5)). - Vaclav Kotesovec, Feb 04 2014