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%I #51 Mar 15 2024 21:28:12
%S 1,3,21,183,1785,18651,204141,2310447,26819121,317530227,3819724293,
%T 46553474919,573608632233,7133530172619,89423593269213,
%U 1128765846337887,14334721079385441,183021615646831587,2347944226115977461,30250309354902101271,391241497991342192985
%N Series reversion of x*(1-4x)/(1-x) is x*A(x) where A(x) is the generating function.
%C The Hankel transform of this sequence is 12^C(n+1,2).
%C Number of Dyck n-paths with two colors of up (U,u) and two colors of down (D,d) avoiding UD. - _David Scambler_, Jun 24 2013
%C Number of small Schröder n-paths with 3 types of up steps (i.e., lattice paths from (0,0) to (2n,0) using steps U1=U2=U3=(1,1), F=(2,0), D=(1,-1), with no F steps on the x-axis). - _Yu Hin Au_, Dec 05 2019
%H J. Abate and W. Whitt, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Whitt/whitt2.html">Integer Sequences from Queueing Theory </a>, J. Int. Seq. 13 (2010), 10.5.5, p_n(3).
%H Yu Hin (Gary) Au, <a href="https://arxiv.org/abs/1912.00555">Some Properties and Combinatorial Implications of Weighted Small Schröder Numbers</a>, arXiv:1912.00555 [math.CO], 2019.
%H Paul Barry, <a href="http://arxiv.org/abs/1312.0583">Embedding structures associated with Riordan arrays and moment matrices</a>, arXiv:1312.0583 [math.CO], 2013.
%H Z. Chen and H. Pan, <a href="http://arxiv.org/abs/1608.02448">Identities involving weighted Catalan-Schroder and Motzkin Paths</a>, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=3, b=4.
%F a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k.
%F a(n) = (3/4)*A103211(n) for n>0.
%F a(n) = -a(n-1)+4*Sum_{i=0..n-1} a(i)*a(n-i-1)), a(0)=1. - _Vladimir Kruchinin_, Mar 30 2015
%F Conjecture: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - _R. J. Mathar_, Aug 16 2015
%F a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - _Peter Luschny_, Jan 08 2018
%F G.f.: (1 + x - sqrt(1 - 14*x + x^2))/(8*x). - _Michael Somos_, Jul 27 2022
%F From _Michael Somos_, Mar 15 2024: (Start)
%F Given g.f. A(x) and y = 2*x*A(-x^2), then y-1/y = (x-1/x)/2.
%F If a(n) := -a(-1-n) for n<0, then 0 = a(n)*(+a(n+1) -35*a(n+2) +4*a(n+3)) +a(n+1)*(+7*a(n+1) +194*a(n+2) -35*a(n+3)) +a(n+2)*(+7*a(n+2) +a(n+3)) for all n in Z. (End)
%e G.f. = 1 + 3*x + 21*x^2 + 183*x^3 + 1785*x^4 + 18651*x^5 + ... - _Michael Somos_, Jul 27 2022
%t Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Mar 30 2015 *)
%t Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* _Peter Luschny_, Jan 08 2018 *)
%t a[ n_] := SeriesCoefficient[(1 + x - Sqrt[1 - 14*x + x^2])/(8*x), {x, 0, n}]; (* _Michael Somos_, Jul 27 2022 *)
%t a[ n_] := (-1)^n * Hypergeometric2F1[ -n, n+1, 2, 4]; (* _Michael Somos_, Mar 15 2024 *)
%o (PARI) Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ _Michel Marcus_, Mar 30 2015
%o (PARI) {a(n) = if(n<0, 0, n++; polcoeff(serreverse(x*(1-4*x)/(1-x) + x*O(x^n)), n))}; /* _Michael Somos_, Jul 27 2022 */
%o (PARI) {a(n) = if(n<0, -a(-1-n), polcoeff(2/(1 + x + sqrt(1 - 14*x + x^2 + x*O(x^n))), n))}; /* _Michael Somos_, Mar 15 2024 */
%Y Cf. A086810, A103211.
%K nonn
%O 0,2
%A _Philippe Deléham_, Oct 29 2007, Nov 06 2007
%E a(17) corrected by _Mark van Hoeij_, Jul 01 2010
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