A188314 - OEIS

%I #17 Sep 08 2022 08:45:56

%S 1,2,5,16,57,219,883,3687,15803,69128,307363,1385003,6310869,29028616,

%T 134610771,628612921,2953640371,13953726888,66240021987,315812059436,

%U 1511569447859,7260364084997,34984937594741,169073568381936,819288294835939,3979892232651125,19377475499900015

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

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

%H G. C. Greubel, <a href="/A188314/b188314.txt">Table of n, a(n) for n = 0..1400</a>

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

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

%F 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)}}.

%F 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

%F G.f. A(x) satisfies: A(x) = 1 + x * (1 + x*A(x) + A(x)^2). - _Ilya Gutkovskiy_, Jul 01 2020

%t 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 *)

%o (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

%o (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

%Y Cf. A000108, A188312.

%K nonn,easy

%O 0,2

%A _Paul Barry_, Mar 28 2011