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%I A200312 #33 Apr 12 2023 11:08:00
%S A200312 1,1,8,35,266,1680,12804,93093,726440,5635058,45063668,362121760,
%T A200312 2955642508,24284658100,201428123040,1680921310635,14119413718770,
%U A200312 119205791509200,1011387051005100,8617021562542470,73704123363739440,632601537174078420
%N A200312 a(n) = A000108(n)*A006130(n), where A000108 is the Catalan numbers and A006130(n) = A006130(n-1) + 3*A006130(n-2).
%C A200312 More generally, given {S} such that: S(n) = b*S(n-1) + c*S(n-2), S(0)=1, |b|>0, |c|>0, then Sum_{n>=0} S(n)*Catalan(n)*x^n = sqrt( (1-2*b*x - sqrt(1-4*b*x-16*c*x^2))/(2*b^2+8*c) )/x.
%H A200312 G. C. Greubel, <a href="/A200312/b200312.txt">Table of n, a(n) for n = 0..450</a>
%H A200312 S. B. Ekhad and M. Yang, <a href="http://sites.math.rutgers.edu/~zeilberg/tokhniot/oMathar1maple12.txt">Proofs of Linear Recurrences of Coefficients of Certain Algebraic Formal Power Series Conjectured in the On-Line Encyclopedia Of Integer Sequences</a>, (2017)
%F A200312 G.f.: sqrt( (1-2*x - sqrt(1-4*x-48*x^2))/26 )/x.
%F A200312 G.f.: (1/x)*Series_Reversion( x*sqrt(1-12*x^2) - x^2 ).
%F A200312 G.f.: (1/x)*Series_Reversion( x-x^2 - 6*x^3*Sum_{n>=0} A000108(n)*3^n*x^(2*n) ).
%F A200312 G.f. satisfies: A(x) = sqrt(1 + 2*x*A(x)^2 + 13*x^2*A(x)^4).
%F A200312 Conjecture: n*(n+1)*a(n) -2*n*(2*n-1)*a(n-1) -12*(2*n-1)*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 17 2011
%F A200312 a(n) = ( ((1+sqrt(13))/2)^(n+1) - ((1-sqrt(13))/2)^(n+1) )/sqrt(13) * binomial(2*n+1,n)/(2*n+1). - _Paul D. Hanna_, Sep 25 2012
%F A200312 0 = +a(n)*(+110592*a(n+3) -9216*a(n+4) -7392*a(n+5) +858*a(n+6)) +a(n+1)*(+6912*a(n+3) -1968*a(n+4) -910*a(n+5) +154*a(n+6)) +a(n+2)*(-240*a(n+3) -2*a(n+4) +41*a(n+5) -4*a(n+6)) +a(n+3)*(+6*a(n+3) +5*a(n+4) +3*a(n+5) -a(n+6)) for all n in Z. - _Michael Somos_, Jul 28 2018
%e A200312 G.f.: A(x) = 1 + x + 2*4*x^2 + 5*7*x^3 + 14*19*x^4 + 42*40*x^5 + 132*97*x^6 + 429*217*x^7 + ... + A000108(n)*A006130(n)*x^n + ...
%e A200312 where the g.f. of A006130, F(x) = 1/(1-x-3*x^2), begins:
%e A200312 F(x) = 1 + x + 4*x^2 + 7*x^3 + 19*x^4 + 40*x^5 + 97*x^6 + 217*x^7 + ...
%t A200312 CoefficientList[Series[Sqrt[(1 - 2*x - Sqrt[1 - 4*x - 48*x^2])/26]/x, {x, 0, 30}], x] (* _G. C. Greubel_, Jul 27 2018 *)
%o A200312 (PARI) {a(n)=binomial(2*n, n)/(n+1)*polcoeff(1/(1-x-3*x^2+x*O(x^n)),n)}
%o A200312 (PARI) {a(n)=polcoeff(sqrt((1-2*x - sqrt(1-4*x-48*x^2+x^3*O(x^n)))/26)/x,n)}
%o A200312 (PARI) {a(n)=polcoeff(serreverse(x*sqrt(1-12*x^2+x^2*O(x^n)) - x^2)/x,n)}
%o A200312 (PARI) {a(n)=polcoeff((1/x)*serreverse(x-x^2 - 6*x^3*sum(m=0,n\2,binomial(2*m,m)/(m+1)*3^m*x^(2*m))+x^3*O(x^n)),n)}
%o A200312 (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!(Sqrt((1-2*x - Sqrt(1-4*x-48*x^2))/26)/x)); // _G. C. Greubel_, Jul 27 2018
%Y A200312 Cf. A098614, A098616, A200375.
%K A200312 nonn
%O A200312 0,3
%A A200312 _Paul D. Hanna_, Nov 16 2011

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