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

0,2

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

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

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 *)

Prog

(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
(Magma) m:=25; R<x>:=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

Crossrefs

Cf. A000108, A188312.

Sequence in context: A072110 A323229 A197158 * A114296 A121689 A357580

Adjacent sequences: A188311 A188312 A188313 * A188315 A188316 A188317

Keyword

nonn,easy

Author

Paul Barry, Mar 28 2011

Status

approved