A098469 A sequence related to the even-indexed Catalan numbers.

1, 2, 6, 20, 78, 332, 1516, 7240, 35734, 180620, 929940, 4858328, 25687052, 137177016, 738819672, 4008435984, 21886788582, 120178329740, 663179894788, 3675923244856, 20456707469540, 114254175491304, 640223315385576

Offset

0,2

Comments

Binomial transform of A098465. Second binomial transform of (1,0,2,0,14,0,132,0,1430,...) (set odd-indexed Catalan numbers to zero).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

G.f.: (sqrt(1+2*x) - sqrt(1-6*x))/(4*x*sqrt(1-2*x)).

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

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

Recurrence: n*(n+1)*a(n) = 4*n*(2*n-1)*a(n-1) - 4*(2*n^2 - 4*n + 3)*a(n-2) - 16*(n-2)*(2*n-3)*a(n-3) + 48*(n-3)*(n-2)*a(n-4). - Vaclav Kotesovec, Oct 24 2012

a(n) ~ 3*6^(n+1/2)/(8*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 24 2012

Mathematica

CoefficientList[Series[(Sqrt[1+2*x]-Sqrt[1-6*x])/(4*x*Sqrt[1-2*x]), {x, 0, 20}], x] (* Vaclav Kotesovec, Oct 24 2012 *)

Prog

(PARI) x='x+O('x^66); Vec((sqrt(1+2*x)-sqrt(1-6*x))/(4*x*sqrt(1-2*x))) \\ Joerg Arndt, May 11 2013

Crossrefs

Cf. A048990.

Sequence in context: A370947 A150181 A150182 * A081563 A038393 A357798

Adjacent sequences: A098466 A098467 A098468 * A098470 A098471 A098472

Keyword

easy,nonn

Author

Paul Barry, Sep 09 2004, corrected Mar 31 2007

Status

approved