A103972 Expansion of (1-sqrt(1-4*x-20*x^2))/(2*x).

1, 6, 12, 60, 264, 1392, 7392, 41424, 236640, 1384512, 8224896, 49554816, 301884672, 1856878080, 11514915840, 71915838720, 451938731520, 2855705994240, 18132621772800, 115637702461440, 740356410961920, 4756888756101120, 30662391191715840, 198229520200704000, 1285001080928845824

Offset: 0

Author

Paul Barry, Feb 23 2005

Keywords

• easy
• nonn

Comments

Image of c(x), the g.f. of the Catalan numbers A000108 under the mapping g(x)->(1+5x)g(x(1+5x)). In general, the image of the Catalan numbers under the mapping g(x)->(1+i*x)g(x(1+i*x)) is given by a(n)=sum{k=0..n, i^(n-k)C(k)C(k+1,n-k)}.

More generally, the sequence C for which C(0)=a, C(1)=b and C(n+1)=sum(C(k)*C(n-k),k=0..n) has the following G.f f: f(z)= (1-sqrt(1-4*z*(a-(a^2-b)*z)))/(2*z). We obtain: C(n)=(sum(-1)^(p-1)*2^{n-p}a^{n-2*p-1}*(a^2-b)^p*((2*n-2*p-1)*...*5*3*1/(p!*(n-2*p+1)!)),p=0..floor((n+1)/2)). By following L. Comtet [Analyse Combinatoire Tomes 1 et 2, PUF, Paris 1970], we obtain also: (n+1)*C(n)-2*a*(2*n-1)*C(n-1)+4*(n-2)*(a^2-b)*C(n-2)=0. - Richard Choulet, Dec 17 2009

Links

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

Crossrefs

Cf. A000108, A025227, A025228, A025229, A025230, A025231, A025232. [From Richard Choulet, Dec 17 2009]

Programs

Maple

n:=30:a(0):=1:a(1):=6 :for k from 1 to n do a(k+1):=sum('a(p)*a(k-p)','p'=0..k):od:seq(a(k),k=0..n); # Richard Choulet, Dec 17 2009

Mathematica

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

PARI

x='x+O('x^66); Vec((1-sqrt(1-4*x-20*x^2))/(2*x)) \\ Joerg Arndt, May 13 2013

Formula

G.f.: (1-sqrt(1-4*x*(1+5*x)))/(2*x).

a(n) = Sum_{k=0..n} 5^(n-k)*C(k)*C(k+1, n-k).

Another recurrence formula: (n+1)*a(n)=2*(2n-1)*a(n-1)+20*(n-2)*a(n-2). - Richard Choulet, Dec 17 2009

a(n) ~ sqrt(12+2*sqrt(6))*(2+2*sqrt(6))^n/(2*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 17 2012