A188314 Expansion of (1/(1-x))*c(x/((1-x)*(1-x^2))) where c(x) is the g.f. of A000108.

1, 2, 5, 16, 57, 219, 883, 3687, 15803, 69128, 307363, 1385003, 6310869, 29028616, 134610771, 628612921, 2953640371, 13953726888, 66240021987, 315812059436, 1511569447859, 7260364084997, 34984937594741, 169073568381936, 819288294835939, 3979892232651125, 19377475499900015

Offset: 0

Author

Paul Barry, Mar 28 2011

Keywords

• nonn
• easy

Comments

Hankel transform is the (25,-29) Somos-4 sequence A188315. Image of the Catalan numbers by A060098.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1400

Crossrefs

Cf. A000108, A188312.

Programs

Magma

m:=25; R:=PowerSeriesRing(Integers(), m); Coefficients(R!((1-x^2- Sqrt(1-4*x-6*x^2+x^4))/(2*x))); // G. C. Greubel, Aug 14 2018

Mathematica

CoefficientList[Series[(1-x^2 - Sqrt[1-4*x-6*x^2+x^4])/(2*x), {x, 0, 50}], x] (* G. C. Greubel, Aug 14 2018 *)

PARI

x='x+O('x^30); Vec((1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x)) \\ G. C. Greubel, Aug 14 2018

Formula

G.f.: (1-x^2- sqrt(1-4*x-6*x^2+x^4))/(2*x).

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

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

Conjecture: (n+1)*a(n) +(n+2)*a(n-1) +(42-26*n)*a(n-2) +30*(3-n)*a(n-3) +(n-5)*a(n-4) +5*(n-6)*a(n-5)=0. - R. J. Mathar, Nov 15 2011

G.f. A(x) satisfies: A(x) = 1 + x * (1 + x*A(x) + A(x)^2). - Ilya Gutkovskiy, Jul 01 2020