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A082590 Expansion of 1/((1-2*x)*sqrt(1-4*x)). 21
1, 4, 14, 48, 166, 584, 2092, 7616, 28102, 104824, 394404, 1494240, 5692636, 21785872, 83688344, 322494208, 1246068806, 4825743832, 18726622964, 72798509728, 283443548276, 1105144970992, 4314388905704, 16862208539008, 65972020761116, 258354647959984 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,2

COMMENTS

Row sums of A068555 and A112336. - Paul Barry, Sep 04 2005

Hankel transform is 2^n*(-1)^C(n+1,2) (A120617). [From Paul Barry, Apr 26 2009]

Number of n-lettered words in the alphabet {1, 2, 3, 4} with as many occurrences of the substring (consecutive subword) [1, 2] as of [1, 3]. - N. J. A. Sloane, Apr 08 2012

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 0..300

Shalosh B. Ekhad and Doron Zeilberger, Automatic Solution of Richard Stanley's Amer. Math. Monthly Problem #11610 and ANY Problem of That Type, arXiv preprint arXiv:1112.6207, 2011. See subpages for rigorous derivations of g.f., recurrence, asymptotics for this sequence.

Alejandro Erickson, Frank Ruskey, Enumerating maximal tatmi mat coverings of square grids with v vertical dominoes, arXiv:1304.0070

Y. Kamiyama, On the middle dimensional homology classes of equilateral polygon spaces, arXiv preprint arXiv:1507.03161, 2015

FORMULA

a(n) = 2^n*JacobiP(n, 1/2, -1-n, 3).

A034430(n) = (n!/2^n)*a(n). A076729(n) = n!*a(n).

a(n) = sum{k=0..n+1, binomial(2n+2, k)*sin((n-k+1)*pi/2} - Paul Barry, Nov 02 2004

a(n) = sum{k=0..n, 2^(n-k)*binomial(2k, k)}; a(n)=sum{k=0..n, (2k)!(2(n-k))!/(n!k!(n-k)!)}; - Paul Barry, Sep 04 2005

a(n) = sum{k=0..n, C(2n,n)C(n,k)/C(2n,2k)} - Paul Barry, Mar 18 2007

G.f.: 1/(1-4x+2x^2/(1+x^2/(1-4x+x^2/(1+x^2/(1-4x+x^2/(1+... (continued fraction). - Paul Barry, Apr 26 2009

Conjecture: n*a(n) +2*(-3*n+1)*a(n-1) +4*(2*n-1)*a(n-2)=0. - R. J. Mathar, Dec 03 2012

a(n) ~ 2^(2*n+1)/sqrt(Pi*n). - Vaclav Kotesovec, Aug 15 2013

a(n) = 2^(n+1)*pochhammer(1/2, n+1)*hyper2F1([1/2,-n], [3/2], -1)/n!. - Peter Luschny, Aug 02 2014

MAPLE

A082590 := proc(n)

    coeftayl( 1/(1-2*x)/sqrt(1-4*x), x=0, n) ;

end proc: # R. J. Mathar, Nov 06 2013

MATHEMATICA

CoefficientList[ Series[ 1/((1 - 2*x)*Sqrt[1 - 4*x]), {x, 0, 25}], x] (* Jean-Fran├žois Alcover, Mar 26 2013 *)

Table[2^(n) JacobiP[n, 1/2, -1-n, 3], {n, 0, 30}] (* Vincenzo Librandi, May 26 2013 *)

CROSSREFS

Bisection of A226302.

Sequence in context: A071749 A071753 A071757 * A085280 A007851 A014325

Adjacent sequences:  A082587 A082588 A082589 * A082591 A082592 A082593

KEYWORD

nonn

AUTHOR

Vladeta Jovovic, May 13 2003

STATUS

approved

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Last modified March 21 04:59 EDT 2019. Contains 321364 sequences. (Running on oeis4.)