A082590 Expansion of 1/((1 - 2*x)*sqrt(1 - 4*x)).

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, 1012627828868072

Offset: 0

Author

Vladeta Jovovic, May 13 2003

Keywords

• nonn

Comments

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

Hankel transform is 2^n*(-1)^C(n+1,2) (A120617). - 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:1112.6207 [math.CO], 2011. See subpages for rigorous derivations of the g.f., the recurrence, asymptotics for this sequence.
• Alejandro Erickson and Frank Ruskey, Enumerating maximal tatami mat coverings of square grids with v vertical dominoes, arXiv:1304.0070 [math.CO], 2013.
• Y. Kamiyama, On the middle dimensional homology classes of equilateral polygon spaces, arXiv:1507.03161 [math.AT], 2015.

Crossrefs

Bisection of A226302.

Cf. A034430, A068555, A076729, A112336.

Programs

Maple

A082590 := proc(n)
    coeftayl( 1/(1-2*x)/sqrt(1-4*x),x=0,n) ;
end proc: # R. J. Mathar, Nov 06 2013
A082590 := n -> 2^n*JacobiP(n, 1/2, -1 - n, 3):
seq(simplify(A082590(n)), n = 0..26);  # Peter Luschny, Jan 22 2025

Mathematica

CoefficientList[ Series[ 1/((1 - 2*x)*Sqrt[1 - 4*x]), {x, 0, 25}], x] (* Jean-François Alcover, Mar 26 2013 *)
Table[2^(n) JacobiP[n, 1/2, -1-n, 3], {n, 0, 30}] (* Vincenzo Librandi, May 26 2013 *)

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(2*n+2, k) * sin((n - k + 1)*Pi/2). - Paul Barry, Nov 02 2004

From Paul Barry, Sep 04 2005: (Start)

a(n) = Sum_{k=0..n} 2^(n-k)*binomial(2*k, k).

a(n) = Sum_{k=0..n} (2*k)! * (2*(n-k))!/(n!*k!*(n-k)!). (End)

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

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

D-finite with recurrence: 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

a(n) - 2*a(n-1) = A000984(n). - R. J. Mathar, Apr 24 2024

a(n) = 2^n*JacobiP(n, 1/2, -1 - n, 3). - Peter Luschny, Jan 22 2025