OFFSET
1,3
LINKS
Robert Israel, Table of n, a(n) for n = 1..1109
Valery A. Liskovets and Timothy R. S. Walsh, Enumeration of Eulerian and unicursal planar maps, Discr. Math., 282 (2004), 209-221.
Youngja Park and SeungKyung Park, Enumeration of generalized lattice paths by string types, peaks, and ascents, Discrete Mathematics 339.11 (2016): 2652-2659.
FORMULA
a(n) = 2^(n-2)*(n-2)*binomial(2n-2, n-1)/n, n>1.
From Robert Israel, Nov 12 2016: (Start)
G.f.: 32*x^3/(sqrt(1-8*x)*(1+sqrt(1-8*x))^3).
E.g.f.: ((1-6*x)/4)*exp(4*x)*I_0(4*x)+(3/2)*exp(4*x)*I_1(4*x)+x/2-1/4, where I_0 and I_1 are modified Bessel functions of the first kind.
a(n+1) = (4*(n-1)*(2*n-1)/((n+1)*(n-2)))*a(n).
a(n) ~ 8^n/(16*sqrt(Pi*n)). (End)
From Amiram Eldar, Jan 16 2024: (Start)
Sum_{n>=3} 1/a(n) = 11/14 - 26*arcsin(1/(2*sqrt(2)))/(7*sqrt(7)).
Sum_{n>=3} (-1)^(n+1)/a(n) = 37*log(2)/27 - 13/18. (End)
EXAMPLE
G.f. = 4*x^3 + 40*x^4 + 336*x^5 + 2688*x^6 + 21120*x^7 + 164736*x^8 + ...
MAPLE
0, seq(2^(n-2)*(n-2)*binomial(2*n-2, n-1)/n, n=2..30); # Robert Israel, Nov 12 2016
MATHEMATICA
a[ n_] := SeriesCoefficient[ ((1 - Sqrt[1 - 8 x])/2)^3 / (2 Sqrt[1 - 8 x] ), {x, 0, n}]; (* Michael Somos, Nov 13 2016 *)
PROG
(Magma) [0] cat [2^(n-2)*(n-2)*Binomial(2*n-2, n-1)/n: n in [2..25]]; // Vincenzo Librandi, Nov 13 2016
CROSSREFS
KEYWORD
easy,nonn
AUTHOR
Valery A. Liskovets, Apr 07 2002
STATUS
approved