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%I #55 Oct 27 2021 13:45:02
%S 1,2,10,76,764,9496,140152,2390480,46206736,997313824,23758664096,
%T 618884638912,17492190577600,532985208200576,17411277367391104,
%U 606917269909048576,22481059424730751232,881687990282453393920,36494410645223834692096,1589659519990672490875904
%N Bisection of A000085.
%C Number of tableaux on 2n elements. - _Roberto E. Martinez II_, Jan 09 2002
%C 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 a(n) = A229223(2n,2) = A229243(2,n). - _Alois P. Heinz_, Sep 17 2013
%D 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 Alois P. Heinz, <a href="/A066223/b066223.txt">Table of n, a(n) for n = 0..200</a>
%H 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 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 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 a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - _Benoit Cloitre_, May 01 2003
%F a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - _Vladeta Jovovic_, May 10 2003
%F E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - _Paul Barry_, May 26 2003
%F E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - _Paul Barry_, Apr 12 2010
%F a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - _Paul Barry_, Apr 21 2010
%F 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 Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - _Emanuele Munarini_, Aug 31 2017
%F 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 a:= proc(n) option remember; `if`(n<2, n+1,
%p (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))
%p end:
%p seq(a(n), n=0..20); # _Alois P. Heinz_, Sep 17 2013
%t NumberOfTableaux[2n]
%t 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 Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* _Emanuele Munarini_, Aug 31 2017 *)
%o (PARI) a(n)=sum(k=0,n,binomial(2*n,2*k)*prod(i=1,k,2*i-1))
%o (PARI) a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)),n))
%Y Cf. A066224.
%K nonn,easy
%O 0,2
%A _N. J. A. Sloane_, Dec 19 2001
%E More terms from _Roberto E. Martinez II_, Jan 09 2002
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