A220910 Matchings avoiding the pattern 231.

1, 1, 3, 14, 83, 570, 4318, 35068, 299907, 2668994, 24513578, 230981316, 2222973742, 21777680644, 216603095388, 2182653550712, 22245324259811, 228995136248850, 2378208988952434, 24893925007653748, 262424206657706682, 2784074166633171596, 29707452318776988260, 318664451642694840264

Offset: 0

Author

N. J. A. Sloane, Jan 02 2013

Keywords

• nonn

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.
• Matteo Cervetti and Luca Ferrari, Pattern avoidance in the matching pattern poset, arXiv:2009.01024 [math.CO], 2020.
• 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.

Crossrefs

Cf. A220911.

Programs

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 *)

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

Formula

G.f.: 54*z/(1+36*z-(1-12*z)^(3/2)) [Cervetti-Ferrari]. - N. J. A. Sloane, Nov 30 2020

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

D-finite with recurrence 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

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