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%I #125 Oct 18 2023 12:34:44
%S 1,1,4,17,74,326,1446,6441,28770,128750,576944,2587850,11615932,
%T 52167688,234383146,1053386937,4735393794,21291593238,95747347176,
%U 430624242942,1936925461644,8712882517188,39195738193836,176335080590442,793336332850164,3569368545752076
%N G.f.: (1-2*x*c(x))/(1-3*x*c(x)) where c(x) = (1 - sqrt(1-4*x))/(2*x) is the g.f. for Catalan numbers A000108.
%C Row sums of triangle A035324.
%C a(n+1) = {1, 4, 17, 74, 326, ...} is the binomial transform of A059738. - _Philippe Deléham_, Nov 26 2009
%C (1, 4, 17, 74, 326, ...) is the invert transform of the odd-indexed central binomial coefficients, A001700. - _David Callan_, Oct 14 2012
%C The sequence starting with index 1 is the INVERT transform of A001700: (1, 3, 10, 35, 126, ...) and the second INVERT transform of the Catalan numbers starting with index 1: (1, 2, 5, 14, 42, ...). - _Gary W. Adamson_, Jun 23 2015
%C From _Peter Bala_, Jan 27 2020: (Start)
%C This sequence is the main diagonal of the lower triangular array formed by taking the first column (k = 0) of the array equal to (1,1,3,9,27,...) - powers of 3 with 1 prepended - and then completing the triangle using the relation T(n,k) = T(n-1,k) + T(n,k-1) for k >= 1. See my link in A001517.
%C 1
%C 1 1
%C 3 4 4
%C 9 13 17 17
%C 27 40 57 74 74
%C 81 121 178 252 326 326
%C ...
%C (End)
%D L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
%H Vincenzo Librandi, <a href="/A049027/b049027.txt">Table of n, a(n) for n = 0..200</a>
%H José Agapito, Ângela Mestre, Maria M. Torres, and Pasquale Petrullo, <a href="https://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 and <a href="https://arxiv.org/abs/1505.05568">arXiv version</a>, arXiv:1505.05568 [math.CO], 2015.
%H Paul Barry and Arnauld Mesinga Mwafise, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL21/Barry/barry362.html">Classical and Semi-Classical Orthogonal Polynomials Defined by Riordan Arrays, and Their Moment Sequences</a>, Journal of Integer Sequences, Vol. 21 (2018), Article 18.1.5.
%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.
%H 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)
%H Milan Janjić, <a href="https://www.emis.de/journals/JIS/VOL21/Janjic2/janjic103.html">Pascal Matrices and Restricted Words</a>, J. Int. Seq., Vol. 21 (2018), Article 18.5.2.
%H Wolfdieter Lang, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL3/LANG/lang.html">On generalizations of Stirling number triangles</a>, J. Integer Seqs., Vol. 3 (2000), #00.2.4.
%H J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F G.f.: x*c(x)/(1-3*x*c(x)), c(x)= g.f. of Catalan numbers A000108.
%F a(n+1) = Sum_{k=0..n} 2^k*comb(2n+1, n-k)*2*(k+1)/(n+k+2) - _Paul Barry_, Jun 22 2004
%F a(n) = (9*a(n-1) - Catalan(n-1))/2, n > 1. - _Vladeta Jovovic_, Aug 08 2004
%F a(n+1) = Sum_{k=0..n} A039598(n,k)*2^k. - _Philippe Deléham_, Mar 21 2007
%F G.f.: 2 / (3 - 1 / sqrt(1 - 4*x)). - _Michael Somos_, Apr 08 2007
%F a(n) = Sum_{k=0..n} A039599(n,k)*A001045(k), for n >= 1. - _Philippe Deléham_, Jun 10 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,-3). - _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 4, 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: 2*n*a(n) + (12-17*n)*a(n-1) + 18*(2*n-3)*a(n-2) = 0. - _R. J. Mathar_, Nov 14 2011
%F a(n) ~ 3^(2*n-1)/2^(n+1). - _Vaclav Kotesovec_, Oct 08 2012
%F 0 = a(n)*(1296*a(n+1) - 1098*a(n+2) + 180*a(n+3)) + a(n+1)*(-126*a(n+1) + 253*a(n+2) - 58*a(n+3)) + a(n+2)*(-10*a(n+2) + 4*a(n+3)) if n > 0. - _Michael Somos_, Jan 23 2014
%F O.g.f.: A(x) = 1/(1 - (1/2)*Sum_{n >= 1} binomial(2*n,n)*x^n). - _Peter Bala_, Sep 01 2016
%F a(n) = 3^(2*n-1)/2^(n+1) + 2^n * (2*n-1)!! * hypergeom([1,n+1], [n+2], 8/9)/(9*(n+1)!) + 0^n * 2/3. - _Vladimir Reshetnikov_, Oct 08 2016
%e G.f. = 1 + x + 4*x^2 + 17*x^3 + 74*x^4 + 326*x^5 + 1446*x^6 + 6441*x^7 + ...
%p a:= proc(n) option remember; `if`(n<3, 1+3*n*(n-1)/2,
%p (17/2-6/n)*a(n-1)-(18-27/n)*a(n-2))
%p end:
%p seq(a(n), n=0..28); # _Alois P. Heinz_, Jan 28 2020
%t Table[SeriesCoefficient[2/(3-1/Sqrt[1-4*x]),{x,0,n}],{n,0,20}] (* _Vaclav Kotesovec_, Oct 08 2012 *)
%t FunctionExpand@Table[3^(2n-1)/2^(n+1) + 2^n (2n-1)!! Hypergeometric2F1[1, n + 1/2, n + 2, 8/9]/(9 (n + 1)!) + 2 KroneckerDelta[n]/3, {n, 0, 20}] (* _Vladimir Reshetnikov_, Oct 08 2016 *)
%o (PARI) {a(n) = if( n<1, n==0, polcoeff( serreverse( x * (1 + 2*x) / (1 + 3*x)^2 + x * O(x^n) ), n))}; /* _Michael Somos_, Apr 08 2007 */
%o (PARI) {a(n) = if( n<0, 0, polcoeff( 2 / (3 - 1 / sqrt(1 - 4*x + x * O(x^n))), n))}; /* _Michael Somos_, Apr 08 2007 */
%o (Magma) [1] cat [n eq 1 select 1 else (9*Self(n-1)-Catalan(n-1))/2: n in [1..30]]; // _Vincenzo Librandi_, Jun 25 2015
%o (Sage) (2/(3-1/sqrt(1-4*x))).series(x, 30).coefficients(x, sparse=False) # _G. C. Greubel_, May 02 2019
%Y Cf. A000108, A001700.
%K easy,nonn
%O 0,3
%A _Wolfdieter Lang_
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