A091712 a(n) = 6*(2n-4)!/((n-2)!n!), if n>2. a(0) = 1, a(1) = a(2) = 2.

1, 2, 2, 2, 3, 6, 14, 36, 99, 286, 858, 2652, 8398, 27132, 89148, 297160, 1002915, 3421710, 11785890, 40940460, 143291610, 504932340, 1790214660, 6382504440, 22870640910, 82334307276, 297670187844, 1080432533656, 3935861372604, 14386251913656, 52749590350072

Offset

0,2

Links

Table of n, a(n) for n=0..30.

Formula

a(n+2) = A007054(n), if n>0.

G.f.: ((1+10x-2x^2)+(1-4x)^(3/2))/2.

a(n) = 6(2n-4)!/((n-2)!n!), if n>2. a(n) = a(n-1)(4n-10)/n, if n>3.

G.f.: A(x) = (2c(x)-1)^3/c(x)^4 = (1-c(x)x)(1+c(x)x)^3, where c(x) = g.f. for Catalan numbers A000108.

From Amiram Eldar, Sep 25 2025: (Start)

a(n) ~ 3 * 2^(2*n-3) / (n^(5/2) * sqrt(Pi)).

Sum_{n>=0} 1/a(n) = 8/3 + 20*Pi/(81*sqrt(3)).

Sum_{n>=0} (-1)^n/a(n) = 59/75 - 8*log(phi)/(25*sqrt(5)), where phi is the golden ratio (A001622). (End)

Mathematica

a[n_] := 6*(2*n - 4)!/((n - 2)!*n!); a[0] = 1; a[1] = a[2] = 2; Array[a, 35, 0] (* Amiram Eldar, Sep 25 2025 *)

Prog

(PARI) a(n)=if(n<3, (n>=0)+(n>0), 6*(2*n-4)!/n!/(n-2)!)
(PARI) a(n)=if(n<0, 0, polcoeff(((1+10*x-2*x^2)+(1-4*x)*sqrt(1-4*x+x*O(x^n)))/2, n))
(PARI) a(n)=if(n<=0, n==0, polcoeff(subst((1-x)*(1+x)^3, x, serreverse(x-x^2+x*O(x^n))), n))

Crossrefs

Essentially the same as A007054.

Cf. A000108, A001622.

Sequence in context: A143596 A342763 A390133 * A125721 A049798 A165198

Adjacent sequences: A091709 A091710 A091711 * A091713 A091714 A091715

Keyword

nonn,easy

Author

Michael Somos, Jan 31 2004

Status

approved