A025229 a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 3, with initial terms 1,3.

1, 3, 6, 21, 78, 318, 1356, 5997, 27222, 126138, 594132, 2836290, 13692300, 66729180, 327855768, 1622216829, 8076311142, 40427919714, 203353800324, 1027318915254, 5210182030308, 26517609163812, 135397544040744, 693364054299474

Offset: 1

Author

Clark Kimberling

Keywords

• nonn
• easy

Links

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

Programs

Maple

A025229 := proc(n)
    option remember;
    if n <=1 then
        n;
    elif n = 2 then
        3;
    else
        add( procname(n-i)*procname(i),i=1..n-1) ;
    end if;
end proc: # R. J. Mathar, Jun 17 2015

Mathematica

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

PARI

a(n)=polcoeff((1-sqrt(1-4*x-8*x^2+x*O(x^n)))/2,n)

Formula

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

a(n) = Sum_{k=0..n} 2^(n-k)*C(k)*C(k+1, n-k) [offset 0]. - Paul Barry, Feb 22 2005

Another recurrence formula: n*a(n) = (4*n-6)*a(n-1)+(8*n-24)*a(n-2). - Richard Choulet, Dec 16 2009

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

Extensions

Name clarified by Robert C. Lyons, Feb 06 2025