A066223 Bisection of A000085.

1, 2, 10, 76, 764, 9496, 140152, 2390480, 46206736, 997313824, 23758664096, 618884638912, 17492190577600, 532985208200576, 17411277367391104, 606917269909048576, 22481059424730751232, 881687990282453393920, 36494410645223834692096, 1589659519990672490875904

Offset: 0

Author

N. J. A. Sloane, Dec 19 2001

Keywords

• nonn
• easy

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
• T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras
• I. Dolinka, J. East and R. D. Gray, Motzkin monoids and partial Brauer monoids, arXiv preprint arXiv:1512.02279 [math.GR], 2015.
• Michael Torpey, Semigroup congruences: computational techniques and theoretical applications, Ph.D. Thesis, University of St. Andrews (Scotland, 2019).

Crossrefs

Cf. A066224.

Programs

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 *)

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))

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

Extensions

More terms from Roberto E. Martinez II, Jan 09 2002