cp's OEIS Frontend

A066223 Bisection of A000085.

%I A066223 #55 Oct 27 2021 13:45:02
%S A066223 1,2,10,76,764,9496,140152,2390480,46206736,997313824,23758664096,
%T A066223 618884638912,17492190577600,532985208200576,17411277367391104,
%U A066223 606917269909048576,22481059424730751232,881687990282453393920,36494410645223834692096,1589659519990672490875904
%N A066223 Bisection of A000085.
%C A066223 Number of tableaux on 2n elements. - _Roberto E. Martinez II_, Jan 09 2002
%C A066223 a(n) = number of ways to connect 2n points labeled 1,2,...,2n in a line with 0 or more arcs such that at most one arc leaves each point. For example, with arcs separated by dashes, a(2)=10 counts {} (no arcs), 12, 13, 14, 23, 24, 34, 12-34, 13-24, 14-23. - _David Callan_, Sep 18 2007
%C A066223 a(n) = A229223(2n,2) = A229243(2,n). - _Alois P. Heinz_, Sep 17 2013
%D A066223 S. Chowla, The asymptotic behavior of solutions of difference equations, in Proceedings of the International Congress of Mathematicians (Cambridge, MA, 1950), Vol. I, 377, Amer. Math. Soc., Providence, RI, 1952.
%H A066223 Alois P. Heinz, <a href="/A066223/b066223.txt">Table of n, a(n) for n = 0..200</a>
%H A066223 T. Copeland, <a href="http://tcjpn.wordpress.com/2012/11/29/infinigens-the-pascal-pyramid-and-the-witt-and-virasoro-algebras/">Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras</a>
%H A066223 I. Dolinka, J. East and R. D. Gray, <a href="http://arxiv.org/abs/1512.02279">Motzkin monoids and partial Brauer monoids</a>, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
%H A066223 Michael Torpey, <a href="https://doi.org/10.17630/10023-17350">Semigroup congruences: computational techniques and theoretical applications</a>, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).
%F A066223 a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - _Benoit Cloitre_, May 01 2003
%F A066223 a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - _Vladeta Jovovic_, May 10 2003
%F A066223 E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - _Paul Barry_, May 26 2003
%F A066223 E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - _Paul Barry_, Apr 12 2010
%F A066223 a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - _Paul Barry_, Apr 21 2010
%F A066223 Conjecture: a(n) +2*(-2*n+1)*a(n-1) +2*(n-1)*(2*n-3)*a(n-2)=0. - _R. J. Mathar_, Nov 24 2012
%F A066223 Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - _Emanuele Munarini_, Aug 31 2017
%F A066223 a(n) ~ n^n*2^(n-1/2)*exp(-n+sqrt(2*n)-1/4) * (1 + 7/(24*sqrt(2*n))). - _Vaclav Kotesovec_, Jun 22 2013
%p A066223 a:= proc(n) option remember; `if`(n<2, n+1,
%p A066223       (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
%p A066223     end:
%p A066223 seq(a(n), n=0..20);  # _Alois P. Heinz_, Sep 17 2013
%t A066223 NumberOfTableaux[2n]
%t A066223 a[n_] := a[n] = If[n<2, n+1, (4*n-2)*a[n-1] - 2*(n-1)*(2*n-3)*a[n-2]]; Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, Oct 13 2014, after _Alois P. Heinz_ *)
%t A066223 Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* _Emanuele Munarini_, Aug 31 2017 *)
%o A066223 (PARI) a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))
%o A066223 (PARI) a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))
%Y A066223 Cf. A066224.
%K A066223 nonn,easy
%O A066223 0,2
%A A066223 _N. J. A. Sloane_, Dec 19 2001
%E A066223 More terms from _Roberto E. Martinez II_, Jan 09 2002