A087626 Expansion of 2/(1-2x+sqrt(1-4x+4x^3)).

1, 2, 5, 13, 36, 104, 311, 955, 2995, 9553, 30896, 101082, 333946, 1112496, 3732955, 12605029, 42800317, 146046819, 500555447, 1722402303, 5948047169, 20607691517, 71610355540, 249520257106, 871614139396, 3051737703526

Offset: 0

Author

Michael Somos, Sep 16 2003

Keywords

• nonn

Examples

			G.f. = 1 + 2*x + 5*x^2 + 13*x^3 + 36*x^4 + 104*x^5 + 311*x^6 + 955*x^7 + ... - _Michael Somos_, Mar 28 2020
		

Links

• Robert Israel, Table of n, a(n) for n = 0..1762
• Jean-Luc Baril and José Luis Ramírez, Descent distribution on Catalan words avoiding ordered pairs of Relations, arXiv:2302.12741 [math.CO], 2023.

Crossrefs

Cf. A006720, A051138, A056010, A157003.

Programs

Maple

f:= gfun:-rectoproc({(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4), a(0) = 1, a(1) = 2, a(2) = 5, a(3) = 13},a(n),remember):
map(f, [$0..50]); # Robert Israel, Oct 26 2018

Mathematica

CoefficientList[Series[2/(1-2x+Sqrt[1-4x+4x^3]),{x,0,30}],x] (* Harvey P. Dale, Jun 12 2017 *)

PARI

{a(n) = polcoeff(2 / (1 - 2*x + sqrt(1 - 4*x + 4*x^3 + x*O(x^n))), n)};

Formula

G.f.: 2/(1-2x+sqrt(1-4x+4x^3)).

G.f. A(x) satisfies 0 = x^2*(1-x)*A(x)^2 - (1-2*x)*A(x) + 1.

First backwards difference is A056010.

(6+4*n)*a(n)+(-6-4*n)*a(n+1)+(-18-4*n)*a(2+n)+(24+5*n)*a(n+3)+(-6-n)*a(n+4)=0. - Robert Israel, Oct 26 2018

HANKEL transform is A006720(n+2). HANKEL transform with 0 prepended is -A051138.

INVERT transform of A157003. - Michael Somos, Mar 28 2020