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 A220910 Matchings avoiding the pattern 231. 2
 1, 1, 3, 14, 83, 570, 4318, 35068, 299907, 2668994, 24513578, 230981316, 2222973742, 21777680644, 216603095388, 2182653550712, 22245324259811, 228995136248850, 2378208988952434, 24893925007653748, 262424206657706682, 2784074166633171596, 29707452318776988260, 318664451642694840264 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,3 LINKS Vincenzo Librandi, Table of n, a(n) for n = 0..200 Jonathan Bloom and Sergi Elizalde, Pattern avoidance in matchings and partitions, arXiv preprint arXiv:1211.3442 [math.CO], 2012. W. Mlotkowski, K. A. Penson, A Fuss-type family of positive definite sequences, arXiv:1507.07312 [math.PR], 2015, Proposition 4.4. Noam Zeilberger and Alain Giorgetti, A correspondence between rooted planar maps and normal planar lambda terms, Logical Methods in Computer Science, vol. 11 (3:22), 2015, pp. 1-39. FORMULA Special values of the hypergeometric function 2F1, in Maple notation: a(n) = (27/8)*doublefactorial(2*n-1)*6^n*hypergeom([2, n+1/2], [n+3], -3)/(n+2)!, n>0. - Karol A. Penson and Wojciech Mlotkowski, Aug 04 2013 Conjecture: n*a(n) +2*(-4*n+17)*a(n-1) +24*(-2*n+3)*a(n-2)=0. - R. J. Mathar, Aug 04 2013 G.f.: ((1-12*x)^(3/2) + (1+36*x)) / (2*(4*x+1)^2). - Vaclav Kotesovec, Aug 23 2014 a(n) ~ 2^(2*n-7) * 3^(n+3) / (sqrt(Pi) * n^(5/2)). - Vaclav Kotesovec, Aug 23 2014 G.f. A(x) satisifies A(x) = 1 + x*A(x)^2*(2 - G(x*A(x)^2))*G(x*A(x)^2)^2, where G(x) = 1 + x*G(x)^4 is the g.f. of A002293. - Paul D. Hanna, Aug 25 2014 MATHEMATICA CoefficientList[Series[((1-12*x)^(3/2) + (1+36*x)) / (2*(4*x+1)^2), {x, 0, 20}], x] (* Vaclav Kotesovec, Aug 23 2014 *) PROG (PARI) x='x+O('x^50); Vec(((1-12*x)^(3/2)+(1+36*x))/(2*(4*x+1)^2)) \\ Altug Alkan, Nov 25 2015 CROSSREFS Sequence in context: A103467 A215661 A224776 * A121687 A154757 A074535 Adjacent sequences:  A220907 A220908 A220909 * A220911 A220912 A220913 KEYWORD nonn AUTHOR N. J. A. Sloane, Jan 02 2013 EXTENSIONS a(11)-a(23) by Karol A. Penson and Wojciech Mlotkowski, Aug 04 2013 Prepended a(0)=1 from Vaclav Kotesovec, Aug 23 2014 STATUS approved

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