A086622 G.f. A(x) satisfies: A(x) = 1/(1-2*x) + x^2*A(x)^2.

1, 2, 5, 12, 30, 76, 197, 520, 1398, 3820, 10594, 29768, 84620, 243000, 704045, 2055760, 6043750, 17875020, 53148310, 158773320, 476311940, 1434313960, 4333867170, 13135533552, 39924668220, 121661345656, 371612931492

Offset: 0

Author

Paul D. Hanna, Jul 24 2003

Keywords

• nonn

Comments

a(n) is the number of Motzkin paths of length n having no (1,0)-steps at levels 1,3,5,... and having (1,0)-steps of two colors at levels 2,4,6,... . Example: a(3) = 12 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 8 paths of shape HHH, 2 paths of shape HUD, and 2 paths of shape UDH. - Emeric Deutsch, May 02 2011

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Carles Cardó, Growth and density in free magmas, arXiv:2401.07827 [math.CO], 2024. See p. 16.

Crossrefs

Cf. A086620 (table), A086621 (diagonal).

Programs

Maple

A086622 := proc(n)
    option remember;
    if n < 3 then
        op(n+1,[1,2,5]) ;
    else
        4*(-n-1)*procname(n-1) +4*procname(n-2) +4*(2*n-3)*procname(n-3) ;
        -%/(n+2) ;
    end if;
end proc:
seq(A086622(n),n=0..20) ; # R. J. Mathar, Nov 02 2021

Mathematica

CoefficientList[Series[(-1+2*x+Sqrt[1-4*x+8*x^3])/(2*(-x^2+2*x^3)), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)

Formula

Antidiagonal sums of square table A086620.

a(n) = Sum_{k=0..floor(n/2)} C(n-k,k) C(2k,k) 2^(n-2k)/(k+1). - Paul Barry, Nov 13 2004

Hankel transform of a(n) is 1,1,1,....; Hankel transform of a(n+1) is A009531(n+2). - Paul Barry, Nov 06 2007

G.f.: 1/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/(1-x^2/(1-2*x-x^2/..... (continued fraction). - Paul Barry, Dec 21 2008

D-finite with recurrence (n+2)*a(n) +4*(-n-1)*a(n-1) +4*a(n-2) +4*(2*n-3)*a(n-3)=0. - R. J. Mathar, Nov 24 2012

G.f.: (-1+2*x+sqrt(1-4*x+8*x^3))/(2*(-x^2+2*x^3)). - Vaclav Kotesovec, Feb 13 2014

a(n) ~ sqrt(50+22*sqrt(5)) * (sqrt(5)+1)^n / (2 * sqrt(Pi) * n^(3/2)). - Vaclav Kotesovec, Feb 13 2014. Equivalently, a(n) ~ 5^(1/4) * 2^n * phi^(n + 5/2) / (sqrt(Pi) * n^(3/2)), where phi = A001622 is the golden ratio. - Vaclav Kotesovec, Dec 07 2021

a(n) = Sum_{i=0..floor(n/2)}2^(n-2i)*C(i)*binomial(n-i,i), where C(n) is the n-th Catalan number A000108. - José Luis Ramírez Ramírez, Apr 20 2015