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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

T. Copeland, Infinitesimal Generators, the Pascal Pyramid, and the Witt and Virasoro Algebras

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

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-Fran├žois Alcover, Oct 13 2014, after Alois P. Heinz *)

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: A136222 A184356 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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Last modified December 4 19:31 EST 2016. Contains 278755 sequences.