A104629 Expansion of (1-2*x-sqrt(1-4*x))/(x^2 * (1+2*x+sqrt(1-4*x))).

1, 2, 6, 18, 57, 186, 622, 2120, 7338, 25724, 91144, 325878, 1174281, 4260282, 15548694, 57048048, 210295326, 778483932, 2892818244, 10786724388, 40347919626, 151355847012, 569274150156, 2146336125648, 8110508473252

Offset: 0

Author

Paul Barry, Mar 17 2005

Keywords

• easy
• nonn

Comments

Diagonal sums of A039598.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000108, A000957, A039598, A064310.

Partial sums of A122920.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!((1-2*x-Sqrt(1-4*x))/(x^2*(1+2*x+Sqrt(1-4*x))))); // G. C. Greubel, Aug 12 2018

Mathematica

CoefficientList[Series[((1-2x-Sqrt[1-4x])/(1+2x+Sqrt[1-4x]))/x^2,{x,0,30}],x] (* Harvey P. Dale, Jul 23 2016 *)
Table[(1 + Sum[CatalanNumber[n]*(-2)^k, {k,0,n+2}])/(8*(-2)^n), {n,0,30}] (* G. C. Greubel, Aug 12 2018 *)

PARI

x='x+O('x^30); Vec((1-2*x-sqrt(1-4*x))/(x^2*(1+2*x+sqrt(1-4*x)))) \\ G. C. Greubel, Aug 12 2018

PARI

for(n=0,30, print1((1 + sum(k=0,n+2, (-2)^k*binomial(2*k, k)/(k+1)))/(8*(-2)^n), ", ")) \\ G. C. Greubel, Aug 12 2018

Python

from itertools import count, islice
def A104629_gen(): # generator of terms
    a, c = 0, 1
    for n in count(1):
        yield (a:=(c:=c*((n<<2)+2)//(n+2))-a>>1)
A104629_list = list(islice(A104629_gen(),20)) # Chai Wah Wu, Apr 26 2023

Formula

a(n) = A000957(n+3).

a(n) = (1 + Sum_{k=0..n+2} C(k)*(-2)^k)/(8*(-2)^n), where C(n) = Catalan numbers.

D-finite with recurrence: 2*(n+3)*a(n) +(-7*n-9)*a(n-1) +2*(-2*n-3)*a(n-2)=0. - R. J. Mathar, Oct 30 2014 [Verified by Georg Fischer, Apr 27 2023]