A085139 a(n) = Sum_{i=0..n-1} (1 + (-1)^(n-1-i))/2 * Sum_{j=0..i} a(j)*a(i-j) for n > 0, with a(0) = 1.

1, 1, 2, 6, 18, 58, 194, 670, 2370, 8546, 31298, 116102, 435346, 1647418, 6283394, 24130174, 93226242, 362098050, 1413098370, 5538138182, 21788069266, 86016385274, 340655956802, 1353023683486, 5388230857538, 21510345134178

Offset: 0

Author

Mario Catalani (mario.catalani(AT)unito.it), Jun 20 2003

Keywords

• easy
• nonn

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..1603
• Paul Barry, Generalized Catalan recurrences, Riordan arrays, elliptic curves, and orthogonal polynomials, arXiv:1910.00875 [math.CO], 2019.
• Paul Barry, On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

Programs

Mathematica

a[n_] := a[n] = (1/2)Sum[Sum[a[j]a[i -j], {j, 0, i}](1 + (-1)^(n+1+i)), {i, 0, n}]; a[0] = 1; Table[a[n], {n, 0, 10}]
(* Second program: *)
Block[{$MaxExtraPrecision = 1000}, CoefficientList[Series[(1/(2 x)) (1 - x^2 - Sqrt[(1 - x^2)^2 - 4 x (1 - x^2)]), {x, 0, 25}], x] ] (* Michael De Vlieger, Jun 06 2023 *)

Formula

G.f.: (1/(2*x)) * (1 - x^2 - sqrt((1 - x^2)^2 - 4*x*(1 - x^2))).

G.f.: C(x/(1-x^2)) where C(x) is the g.f. of A000108. - Paul Barry, Apr 12 2005

G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x/(1-x^2) (continued fraction); this is a special case of the previous formula. - Joerg Arndt, Mar 18 2011

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k)*C(n-2k) - Sum_{k=0..floor((n-2)/2)} C(n-k-2,k)*C(n-2k-2). - Paul Barry, Nov 30 2008

From Paul Barry, May 27 2009: (Start)

G.f.: 1+x/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-2x-x^2/(1-2x-2x^2/(1-x-2x^2/(1-... (continued fraction).

a(n) = 0^n + Sum_{k=0..floor((n-1)/2)} C(n-k-1,k)*A000108(n-2k). (End)

G.f.: M(F(x)) where F(x) is the g.f. of A000045, M(x) is the g.f. A001006. - Vladimir Kruchinin, Sep 06 2010

G.f. A(x) satisfies: A(x) = 1 + x/(1-x^2) * A(x)^2. - Paul D. Hanna, Jul 04 2018

G.f. A(x) satisfies: Sum_{n>=0} log( (1 - (-1)^n*x)/A(x) )^n / n! = 1. - Paul D. Hanna, Jul 04 2018

a(n) ~ 5^(1/4) * phi^(3*n) / (sqrt(2*Pi) * n^(3/2)), where phi = A001622 = (1 + sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jul 04 2018

Extensions

Name revised slightly by Paul D. Hanna, Jul 04 2018