A025231 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3.

2, 3, 12, 57, 300, 1686, 9912, 60213, 374988, 2381322, 15361896, 100389306, 663180024, 4421490924, 29712558576, 201046204173, 1368578002188, 9366084668802, 64403308499592, 444739795023054, 3082969991029800

Offset

1,1

Links

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Formula

G.f.: (1 - sqrt(1 - 8*x + 4*x^2))/2. - Michael Somos, Jun 08 2000

n*a(n) = (8*n - 12)*a(n - 1) - (4*n - 12)*a(n - 2). [Richard Choulet, Dec 16 2009]

G.f.: 1 + x - G(0); G(k) = 1 + 2*x - 3*x/G(k + 1); (continued fraction, 1-step). - Sergei N. Gladkovskii, Jan 05 2012

a(n) ~ sqrt(4*sqrt(3) - 6)*(4 + 2*sqrt(3))^n/(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 07 2012

G.f.: x + x/W(0), where W(k)= 1 - 2*x - x/W(k+1); (continued fraction). - Sergei N. Gladkovskii, Aug 16 2013 [Edited by Michael Somos, Apr 10 2014]

0 = a(n)*(+16*a(n+1) - 80*a(n+2) + 16*a(n+3)) + a(n+1)*(+16*a(n+1) + 56*a(n+2) - 20*a(n+3)) + a(n+2)*(+4*a(n+2) + a(n+3)) if n>0. - Michael Somos, Apr 10 2014

a(n) = Sum_{k=0..n} C(k+1,n-k)*2^(2*k+1-n)*(-1)^(n-k)*C(2*k,k)/(k+1). - Vladimir Kruchinin, Apr 21 2023

Example

G.f. = 2*x + 3*x^2 + 12*x^3 + 57*x^4 + 300*x^5 + 1686*x^6 + 9912*x^7 + ...

Mathematica

Table[SeriesCoefficient[(1 - Sqrt[1 - 8*x + 4*x^2])/2, {x, 0, n}], {n, 1, 20}] (* Vaclav Kotesovec, Oct 07 2012 *)

Prog

(PARI) a(n)=polcoeff((1-sqrt(1-8*x+4*x^2+x*O(x^n)))/2, n)
(Maxima)
a(n):=sum(binomial(k+1, n-k)*2^(2*k+1-n)*(-1)^(n-k)*binomial(2*k, k)/(k+1), k, 0, n); /* _Vladimir Kruchinin, Apr 21 2023 */

Crossrefs

Essentially the same as A047891.

Sequence in context: A002638 A027072 A083746 * A366454 A094532 A092980

Adjacent sequences: A025228 A025229 A025230 * A025232 A025233 A025234

Keyword

nonn,eigen

Author

Clark Kimberling

Status

approved