A085139 - OEIS

%I #41 Jun 06 2023 17:40:49

%S 1,1,2,6,18,58,194,670,2370,8546,31298,116102,435346,1647418,6283394,

%T 24130174,93226242,362098050,1413098370,5538138182,21788069266,

%U 86016385274,340655956802,1353023683486,5388230857538,21510345134178

%N 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.

%H Michael De Vlieger, <a href="/A085139/b085139.txt">Table of n, a(n) for n = 0..1603</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.

%H Paul Barry, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL26/Barry/barry601.html">On Motzkin-Schröder Paths, Riordan Arrays, and Somos-4 Sequences</a>, J. Int. Seq. (2023) Vol. 26, Art. 23.4.7.

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

%F G.f.: C(x/(1-x^2)) where C(x) is the g.f. of A000108. - _Paul Barry_, Apr 12 2005

%F 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

%F 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

%F From _Paul Barry_, May 27 2009: (Start)

%F 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).

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

%F 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

%F G.f. A(x) satisfies: A(x) = 1 + x/(1-x^2) * A(x)^2. - _Paul D. Hanna_, Jul 04 2018

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

%F 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

%t 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}]

%t (* Second program: *)

%t 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 *)

%K easy,nonn

%O 0,3

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

%E Name revised slightly by _Paul D. Hanna_, Jul 04 2018