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 A066223 Bisection of A000085. 5
 1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS Number of tableaux on 2n elements. - Roberto E. Martinez II, Jan 09 2002 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 a(n) = A229223(2n,2) = A229243(2,n). - Alois P. Heinz, Sep 17 2013 REFERENCES 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. LINKS Alois P. Heinz, Table of n, a(n) for n = 0..200 I Dolinka, J East, RD Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279, 2015. FORMULA a(n) = sum(k=0, n, C(2n, 2*k)*(2k-1)!!). - Benoit Cloitre, May 01 2003 a(n) = n!*2^n*LaguerreL(n, -1/2, -1/2). - Vladeta Jovovic, May 10 2003 E.g.f.: cosh(x)*exp(x^2/2) (with interpolated zeros) - Paul Barry, May 26 2003 E.g.f.: exp(x/(1-2*x))/sqrt(1-2*x). - Paul Barry, Apr 12 2010 a(n) = (1/sqrt(2*pi))*Int((1+x)^(2*n)*exp(-x^2/2),x,-infinity,infinity). - Paul Barry, Apr 21 2010 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 Remark: the above conjectured recurrence is true and can be obtained by the e.g.f. - Emanuele Munarini, Aug 31 2017 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 MAPLE a:= proc(n) option remember; `if`(n<2, n+1,       (4*n-2)*a(n-1)-2*(n-1)*(2*n-3)*a(n-2))     end: seq(a(n), n=0..20);  # Alois P. Heinz, Sep 17 2013 MATHEMATICA NumberOfTableaux[2n] 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 *) Table[(-2)^n HypergeometricU[-n, 1/2, -(1/2)], {n, 0, 90}] (* Emanuele Munarini, Aug 31 2017 *) PROG (PARI) a(n)=sum(k=0, n, binomial(2*n, 2*k)*prod(i=1, k, 2*i-1)) (PARI) a(n)=if(n<0, 0, n*=2; n!*polcoeff(exp(x+x^2/2+x*O(x^n)), n)) CROSSREFS Cf. A066224. Sequence in context: A184356 A292631 A124426 * A088500 A195136 A140763 Adjacent sequences:  A066220 A066221 A066222 * A066224 A066225 A066226 KEYWORD nonn,easy AUTHOR N. J. A. Sloane, Dec 19 2001 EXTENSIONS More terms from Roberto E. Martinez II, Jan 09 2002 STATUS approved

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