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A131763 Series reversion of x*(1-4*x)/(1-x) is x*A(x) where A(x) is the generating function.

%I A131763 #66 May 04 2025 01:57:03
%S A131763 1,3,21,183,1785,18651,204141,2310447,26819121,317530227,3819724293,
%T A131763 46553474919,573608632233,7133530172619,89423593269213,
%U A131763 1128765846337887,14334721079385441,183021615646831587,2347944226115977461,30250309354902101271,391241497991342192985
%N A131763 Series reversion of x*(1-4*x)/(1-x) is x*A(x) where A(x) is the generating function.
%C A131763 The Hankel transform of this sequence is 12^C(n+1,2).
%C A131763 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 A131763 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 A131763 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 A131763 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 A131763 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 A131763 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 A131763 a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k.
%F A131763 a(n) = (3/4)*A103211(n) for n>0.
%F A131763 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 A131763 D-finite with recurrence: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - _R. J. Mathar_, Aug 16 2015
%F A131763 a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - _Peter Luschny_, Jan 08 2018
%F A131763 G.f.: (1 + x - sqrt(1 - 14*x + x^2))/(8*x). - _Michael Somos_, Jul 27 2022
%F A131763 From _Michael Somos_, Mar 15 2024: (Start)
%F A131763 Given g.f. A(x) and y = 2*x*A(-x^2), then y-1/y = (x-1/x)/2.
%F A131763 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 A131763 G.f. = 1 + 3*x + 21*x^2 + 183*x^3 + 1785*x^4 + 18651*x^5 + ... - _Michael Somos_, Jul 27 2022
%t A131763 Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x],x]] (* _Vaclav Kotesovec_, Mar 30 2015 *)
%t A131763 Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* _Peter Luschny_, Jan 08 2018 *)
%t A131763 a[ n_] := SeriesCoefficient[(1 + x - Sqrt[1 - 14*x + x^2])/(8*x), {x, 0, n}]; (* _Michael Somos_, Jul 27 2022 *)
%t A131763 a[ n_] := (-1)^n * Hypergeometric2F1[ -n, n+1, 2, 4]; (* _Michael Somos_, Mar 15 2024 *)
%o A131763 (PARI) Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ _Michel Marcus_, Mar 30 2015
%o A131763 (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 A131763 (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 A131763 Cf. A086810, A103211.
%Y A131763 Cf. for series reversion of x*(1-r*x)/(1-x): A001003 (r=2), A107841 (r=3), this sequence (r=4), A131765 (r=5), A131846 (r=6), A131926 (r=7), A131869 (r=8), A131927 (r=9).
%K A131763 nonn
%O A131763 0,2
%A A131763 _Philippe Deléham_, Oct 29 2007, Nov 06 2007
%E A131763 a(17) corrected by _Mark van Hoeij_, Jul 01 2010