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 A131763 Series reversion of x(1-4x)/(1-x). 3
 1, 3, 21, 183, 1785, 18651, 204141, 2310447, 26819121, 317530227, 3819724293, 46553474919, 573608632233, 7133530172619, 89423593269213, 1128765846337887, 14334721079385441, 183021615646831587, 2347944226115977461, 30250309354902101271, 391241497991342192985 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS The Hankel transform of this sequence is 12^C(n+1,2). 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 LINKS J. Abate, W. Whitt, Integer Sequences from Queueing Theory , J. Int. Seq. 13 (2010), 10.5.5, p_n(3). P. Barry, Embedding structures associated with Riordan arrays and moment matrices, arXiv:1312.0583 [math.CO], 2013. Z. Chen, H. Pan, Identities involving weighted Catalan-Schroder and Motzkin Paths, arXiv:1608.02448 [math.CO], (2016), eq. (1.13), a=3, b=4. FORMULA a(n) = Sum_{0<=k<=n} A086810(n,k)*3^k. a(n) = (3/4)*A103211(n) for n>0. 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 Conjecture: (n+1)*a(n) +7*(-2*n+1)*a(n-1) +(n-2)*a(n-2)=0. - R. J. Mathar, Aug 16 2015 a(n) = (-1)^n*hypergeom([-n, n + 1], [2], 4). - Peter Luschny, Jan 08 2018 MATHEMATICA Rest[CoefficientList[InverseSeries[Series[x*(1-4*x)/(1-x), {x, 0, 20}], x], x]] (* Vaclav Kotesovec, Mar 30 2015 *) Table[(-1)^n Hypergeometric2F1[-n, n + 1, 2, 4], {n, 0, 20}] (* Peter Luschny, Jan 08 2018 *) PROG (PARI) Vec(serreverse(x*(1-4*x)/(1-x)+ O(x^30))) \\ Michel Marcus, Mar 30 2015 CROSSREFS Sequence in context: A192946 A216171 A054879 * A006199 A083063 A012163 Adjacent sequences:  A131760 A131761 A131762 * A131764 A131765 A131766 KEYWORD nonn AUTHOR Philippe Deléham, Oct 29 2007, Nov 06 2007 EXTENSIONS a(17) corrected by Mark van Hoeij, Jul 01 2010 STATUS approved

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Last modified February 21 03:06 EST 2019. Contains 320364 sequences. (Running on oeis4.)