A014431 - OEIS

%I #55 Nov 22 2024 08:33:50

%S 1,2,2,6,14,42,122,382,1206,3922,12914,43190,145950,498170,1714026,

%T 5940014,20712646,72623266,255875298,905477734,3216853294,11469069258,

%U 41023019098,147166210014,529374272470,1908965352434,6899707805522,24991194656022,90698707816766

%N a(1) = 1, a(2) = 2, a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-2)*a(2) for n >= 3.

%H Michael De Vlieger, <a href="/A014431/b014431.txt">Table of n, a(n) for n = 1..1724</a>

%H Paul Barry, <a href="https://arxiv.org/abs/1910.00875">Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials</a>, arXiv:1910.00875 [math.CO], 2019.

%F a(n) = 2*A025235(n-2) for n>=2.

%F G.f.: (1+x-sqrt(1-2*x-7*x^2))/2. - _Michael Somos_, Jun 08 2000

%F a(n) = (A084601(n) - A084601(n-1))/(2*(n-1)) for n > 1. - _Mark van Hoeij_, Jul 02 2010

%F G.f.: x + 2*x^2/G(0) with G(k) = (1 - x - 2*x^2/G(k+1)) (continued fraction). - _Nikolaos Pantelidis_, Dec 16 2022

%F From _Peter Bala_, May 01 2024: (Start)

%F O.g.f.: A(x) = x*S(x/(1 + 2*x)) = 2*x - x*S(- x/(1 - 4*x)), where S(x) = (1 - x - sqrt(1 - 6*x + x^2))/(2*x) is the o.g.f. for the large Schröder numbers A006318.

%F The g.f. satisfies A(x)^2 - (1 + x)*A(x) + x*(1 + 2*x) = 0.

%F A(x) = x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - x*(1 + 2*x)/(1 + x - ...))).

%F A(x) = x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - 2*x/(1 + 2*x - x/(1 - ...))))))). (End)

%F D-finite with recurrence n*a(n) +(-2*n+3)*a(n-1) +7*(-n+3)*a(n-2)=0. - _R. J. Mathar_, Nov 22 2024

%t Rest@ CoefficientList[Series[(1 + x - Sqrt[1 - 2 x - 7 x^2])/2, {x, 0, 27}], x] (* _Michael De Vlieger_, Jan 02 2020 *)

%o (PARI) a(n)=polcoeff((1+x-sqrt(1-2*x-7*x^2+x*O(x^n)))/2,n)

%o (Magma) a:=[1,2]; for n in [3..30] do Append(~a,&+[a[k]*a[n-k]:k in [1..n-2]] ); end for; a; // _Marius A. Burtea_, Jan 02 2020

%Y Cf. A025235, A006318, A084601.

%K nonn,easy

%O 1,2

%A _Clark Kimberling_ and _Wouter Meeussen_

%E Corrected by _T. D. Noe_, Oct 31 2006