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%I #35 Sep 08 2022 08:45:26
%S 1,2,3,6,15,42,126,396,1287,4290,14586,50388,176358,624036,2228700,
%T 8023320,29084535,106073010,388934370,1432916100,5301789570,
%U 19692361260,73398801060,274447690920,1029178840950,3869712441972,14585839204356
%N Self-convolution of A120588, such that a(n) = 3*A120588(n) for n >= 2.
%C For n >= 2, a(n) equals 2^(2n+1) times the coefficient of Pi in 2F1([3/2, n+1], [5/2], -1). - _John M. Campbell_, Jul 17 2011
%H G. C. Greubel, <a href="/A120589/b120589.txt">Table of n, a(n) for n = 0..1000</a>
%H Guo-Niu Han, <a href="/A196265/a196265.pdf">Enumeration of Standard Puzzles</a>, 2011. [Cached copy]
%H Guo-Niu Han, <a href="https://arxiv.org/abs/2006.14070">Enumeration of Standard Puzzles</a>, arXiv:2006.14070 [math.CO], 2020.
%H S. Kitaev, J. Remmel and M. Tiefenbruck, <a href="http://arxiv.org/abs/1201.6243">Marked mesh patterns in 132-avoiding permutations I</a>, arXiv:1201.6243 [math.CO], 2012. - From _N. J. A. Sloane_, May 09 2012
%H Sergey Kitaev, Jeffrey Remmel, and Mark Tiefenbruck, <a href="https://arxiv.org/abs/1302.2274">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, arXiv:1302.2274 [math.CO], 2013.
%H Sergey Kitaev, Jeffrey Remmel, Mark Tiefenbruck, <a href="http://www.emis.de/journals/INTEGERS/papers/p16/p16.Abstract.html">Quadrant Marked Mesh Patterns in 132-Avoiding Permutations II</a>, Electronic Journal of Combinatorial Number Theory, 15 (2015), #A16.
%F a(n) = 3*A000108(n-1) for n >= 2, where A000108 are the Catalan numbers.
%F G.f.: (5 - 2*x - 3*sqrt(1-4*x))/2. - _G. C. Greubel_, Feb 18 2019
%e A(x) = 1 + 2*x + 3*x^2 + 6*x^3 + 15*x^4 + 42*x^5 + 126*x^6 + 396*x^7 + ...
%e A(x)^(1/2) = 1 + x + x^2 + 2*x^3 + 5*x^4 + 14*x^5 + 42*x^6 + 132*x^7 + ...
%p A120589List := proc(m) local A, P, n; A := [1,2,3]; P := [3];
%p for n from 1 to m - 2 do P := ListTools:-PartialSums([op(P), P[-1]]);
%p A := [op(A), P[-1]] od; A end: A120589List(26); # _Peter Luschny_, Mar 26 2022
%t Join[{1,2,3}, Table[3*(2*n)!/n!/(n+1)!, {n,2,40}]]
%t CoefficientList[Series[(5-2x -3Sqrt[1-4x])/2, {x, 0, 30}], x] (* _G. C. Greubel_, Feb 18 2019 *)
%o (PARI) {a(n)=local(A=1+x+x^2+x*O(x^n));for(i=0,n,A=A-3*A+2+x+A^2);polcoeff(A^2,n)}
%o (PARI) my(x='x+O('x^30)); Vec((5-2*x-3*sqrt(1-4*x))/2) \\ _G. C. Greubel_, Feb 18 2019
%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( (5-2*x-3*Sqrt(1-4*x))/2 )); // _G. C. Greubel_, Feb 18 2019
%o (Sage) ((5-2*x-3*sqrt(1-4*x))/2).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, Feb 18 2019
%Y Cf. A120588 (A(x)^(1/2)); A120590-A120607.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 16 2006
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