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%I #52 Oct 18 2023 12:22:32
%S 1,1,5,26,137,726,3858,20532,109361,582782,3106550,16562668,88314634,
%T 470942044,2511443268,13393472616,71428622337,380940866574,
%U 2031641406798,10835261623356,57787472903502,308197667445204,1643712737618748,8766437439778776,46754218658948922
%N Expansion of g.f.: (1-3*x*C)/(1-4*x*C) where C = (1 - sqrt(1-4*x))/(2*x) = g.f. for Catalan numbers A000108.
%C From _Paul Barry_, Sep 23 2009: (Start)
%C The Hankel transform of this sequence is 3n+1 or 1,4,7,10,... (A016777).
%C The Hankel transform of the aeration of this sequence is A016777 doubled, that is, 1,1,4,4,7,7,...
%C In general, the Hankel transform of [x^n](1-r*xc(x))/(1-(r+1)*xc(x)) is rn+1, and that of the corresponding aerated sequence is the doubled sequence of rn+1. (End)
%D L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
%H G. C. Greubel, <a href="/A076025/b076025.txt">Table of n, a(n) for n = 0..1000</a>
%H José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, <a href="http://cs.uwaterloo.ca/journals/JIS/VOL18/Agapito/agapito2.html">On One-Parameter Catalan Arrays</a>, Journal of Integer Sequences, Vol. 18 (2015), Article 15.5.1.
%H Richard Ehrenborg, Gábor Hetyei, and Margaret Readdy, <a href="https://arxiv.org/abs/2310.06288">Catalan-Spitzer permutations</a>, arXiv:2310.06288 [math.CO], 2023. See p. 20.
%F a(n+1) = Sum_{k=0..n} 3^k*binomial(2n+1, n-k)*2*(k+1)/(n+k+2). - _Paul Barry_, Jun 22 2004
%F a(n+1) = Sum_{k=0..n} A039598(n,k)*3^k. - _Philippe Deléham_, Mar 21 2007
%F a(n) = Sum_{k=0..n} A039599(n,k)*A015518(k), for n >= 1. - _Philippe Deléham_, Nov 22 2007
%F Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n+1)=(-1)^n*charpoly(A,-4). - _Milan Janjic_, Jul 08 2010
%F From _Gary W. Adamson_, Jul 25 2011: (Start)
%F a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
%F 5, 1, 0, 0, 0, ...
%F 1, 1, 1, 0, 0, ...
%F 1, 1, 1, 1, 0, ...
%F 1, 1, 1, 1, 1, ...
%F ... (End)
%F D-finite with recurrence: 3*n*a(n) +2*(9-14*n)*a(n-1) +32*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ 2^(4*n-1)/3^(n+1). - _Vaclav Kotesovec_, Dec 09 2013
%F The sequence is the INVERT transform of A049027: (1, 4, 17, 74, 326, ...) and the third INVERT transform of the Catalan sequence (1, 2, 5, ...). - _Gary W. Adamson_, Jun 23 2015
%F O.g.f.: A(x) = (1 - 1/2*Sum_{n >= 1} binomial(2*n,n)*x^n)/(1 - Sum_{n >= 1} binomial(2*n,n)*x^n). - _Peter Bala_, Sep 01 2016
%t CoefficientList[Series[(1-3*Sqrt[1-4*x])/(2-4*Sqrt[1-4*x]),{x,0,30}],x] (* _Vaclav Kotesovec_, Dec 09 2013 *)
%t Flatten[{1,Table[FullSimplify[(2*n)! * Hypergeometric2F1Regularized[1, n+1/2, n+2, 3/4] / (16*n!) + 2^(4*n-1)/3^(n+1)], {n,1,30}]}] (* _Vaclav Kotesovec_, Dec 09 2013 *)
%o (PARI) my(x='x+O('x^30)); Vec((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))) \\ _G. C. Greubel_, May 04 2019
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( (1- 3*Sqrt(1-4*x))/(2-4*Sqrt(1-4*x)) )); // _G. C. Greubel_, May 04 2019
%o (Sage) ((1-3*sqrt(1-4*x))/(2-4*sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 04 2019
%Y Cf. A000108, A001700, A049027, A076026.
%K nonn,easy
%O 0,3
%A _N. J. A. Sloane_, Oct 29 2002
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