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A130655 Catalan transform of Catalan numbers C(n+1).

%I A130655 #19 Mar 22 2017 03:35:51
%S A130655 1,2,7,28,119,524,2363,10844,50446,237280,1126437,5389916,25967972,
%T A130655 125868952,613385075,3003586196,14771851093,72936101780,361419276386,
%U A130655 1796837068400,8960207761500
%N A130655 Catalan transform of Catalan numbers C(n+1).
%H A130655 G. C. Greubel, <a href="/A130655/b130655.txt">Table of n, a(n) for n = 0..1000</a>
%F A130655 a(n) = Sum_{k=0..n} A106566(n,k)*A000108(k+1).
%F A130655 Conjecture: 3*n*(n-2)*(n+2)*a(n) +4*(-10*n^3+21*n^2+7*n-15)*a(n-1) +16*(11*n^3-47*n^2+57*n-15)*a(n-2) -8*(2*n-5)*(4*n-9)*(4*n-7)*a(n-3)=0. - _R. J. Mathar_, Mar 01 2015
%F A130655 G.f.: (C(x*C(x))-1)/(x*C(x)), where C(x) is g.f. of Catalan numbers A000108. - _Vladimir Kruchinin_, Jul 02 2015
%F A130655 a(n) ~ 2^(4*n+3/2) / (sqrt(Pi) * n^(3/2) * 3^(n-1/2)). - _Vaclav Kotesovec_, Jul 02 2015
%p A130655 A130655 := proc(n)
%p A130655     add(A106566(n,k)*A000108(k+1),k=0..n) ;
%p A130655 end proc: # _R. J. Mathar_, Mar 01 2015
%t A130655 CoefficientList[Series[2/(Sqrt[-1 + 2*Sqrt[1-4*x]] + Sqrt[1-4*x]),{x,0,20}],x] (* _Vaclav Kotesovec_, Jul 02 2015 *)
%o A130655 (PARI) x='x+O('x^50); Vec(2/(sqrt(-1 + 2*sqrt(1-4*x)) + sqrt(1-4*x))) \\ _G. C. Greubel_, Mar 21 2017
%Y A130655 Cf. A127632, A129442.
%K A130655 nonn
%O A130655 0,2
%A A130655 _Philippe Deléham_, Jun 21 2007