A080243 Signed super-Catalan or little Schroeder numbers.

1, -1, 3, -11, 45, -197, 903, -4279, 20793, -103049, 518859, -2646723, 13648869, -71039373, 372693519, -1968801519, 10463578353, -55909013009, 300159426963, -1618362158587, 8759309660445, -47574827600981, 259215937709463, -1416461675464871, 7760733824437545, -42624971294485657

Offset

0,3

Comments

Row sums of triangle A080245.

References

I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, Wiley, N.Y., 1983, Exercise 2.7.12.(b).

Links

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Formula

G.f.: (-1+x+sqrt(1+6*x+x^2))/x/4. - Vladeta Jovovic, Sep 27 2003

Conjecture: (n+1)*a(n) +3*(2*n-1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Nov 26 2012

G.f.: 1 - x/(Q(0) + x) where Q(k) = 1 + k*(1+x) + x + x*(k+1)*(k+2)/Q(k+1); (continued fraction). - Sergei N. Gladkovskii, Mar 14 2013

a(n) ~ (-1)^n * sqrt(4+3*sqrt(2)) * (3+2*sqrt(2))^n /(4*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Aug 15 2013

G.f. A(x) satisfies: A(x) = (1 - 2*x*A(x)^2) / (1 - x). - Ilya Gutkovskiy, Jun 30 2020

Mathematica

CoefficientList[Series[(-1 + x + Sqrt[1 + 6 x + x^2]) /x / 4, {x, 0, 30}], x] (* Vincenzo Librandi, Aug 05 2013 *)

Prog

(PARI) x='x+O('x^66); Vec( (-1+x+sqrt(1+6*x+x^2))/x/4 ) \\ Joerg Arndt, Aug 15 2013

Crossrefs

Cf. A001003, A080245.

Sequence in context: A217889 A217890 A001003 * A151131 A151132 A200075

Adjacent sequences: A080240 A080241 A080242 * A080244 A080245 A080246

Keyword

sign

Author

Paul Barry, Feb 13 2003

Status

approved