A014431 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.

1, 2, 2, 6, 14, 42, 122, 382, 1206, 3922, 12914, 43190, 145950, 498170, 1714026, 5940014, 20712646, 72623266, 255875298, 905477734, 3216853294, 11469069258, 41023019098, 147166210014, 529374272470, 1908965352434, 6899707805522, 24991194656022, 90698707816766

Offset: 1

Author

Clark Kimberling and Wouter Meeussen

Keywords

• nonn
• easy

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..1724
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.

Crossrefs

Cf. A025235, A006318, A084601.

Programs

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

Mathematica

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

PARI

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

Formula

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

G.f.: (1+x-sqrt(1-2*x-7*x^2))/2. - Michael Somos, Jun 08 2000

a(n) = (A084601(n) - A084601(n-1))/(2*(n-1)) for n > 1. - Mark van Hoeij, Jul 02 2010

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

From Peter Bala, May 01 2024: (Start)

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.

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

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

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)

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

Extensions

Corrected by T. D. Noe, Oct 31 2006