A115962 Expansion of 1/sqrt(1-4*x^2-4*x^3).

1, 0, 2, 2, 6, 12, 26, 60, 130, 300, 672, 1540, 3514, 8064, 18552, 42756, 98802, 228624, 530024, 1230372, 2860000, 6655792, 15505932, 36159552, 84398626, 197154984, 460903796, 1078251044, 2524144224, 5912535672, 13857378300, 32495267712

Offset: 0

Author

Paul Barry, Mar 14 2006

Keywords

• easy
• nonn

Comments

Diagonal sums of number triangle A115951.

Number of lattice paths from (0,0) to (n,n) using steps (2,1), (1,0), (1,2). - Joerg Arndt, Jul 05 2011

Diagonal of rational function 1/(1 - (x^2 + y^2 + x^3*y)). - Seiichi Manyama, Mar 22 2023

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000

Crossrefs

CF. A006139, A361726.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 1/Sqrt(1-4*x^2-4*x^3) )); // G. C. Greubel, May 06 2019

Maple

A115962 := proc(n)
    option remember;
    if n < 4 then
        op(n+1,[1,0,2,2]);
    else
        4*(n-1)*procname(n-2)+2*(2*n-3)*procname(n-3) ;
        %/n ;
    end if;
end proc:
seq(A115962(n),n=0..20) ; # R. J. Mathar, Jan 14 2020

Mathematica

CoefficientList[Series[1/Sqrt[1-4x^2-4x^3], {x, 0, 35}], x] (* or *) Table[Sum[Binomial[2k, k] Binomial[k, n-2k], {k, 0, Floor[n/2]}], {n, 0, 35}] (* Michael De Vlieger, Sep 03 2015 *)

PARI

x = xx+O(xx^40); Vec(1/sqrt(1-4*x^2-4*x^3)) \\ Michel Marcus, Sep 03 2015

Sage

(1/sqrt(1-4*x^2-4*x^3)).series(x, 30).coefficients(x, sparse=False) # G. C. Greubel, May 06 2019

Formula

a(n) = Sum_{k=0..floor(n/2)} C(2*k,k)*C(k,n-2*k).

G.f.: Q(0), where Q(k) = 1 + 4*x*(x+x^2)*(4*k+1) / (4*k+2 - 4*x*(x+x^2)*(4*k+2)*(4*k+3) / (4*x*(x+x^2)*(4*k+3) + 4*(k+1) / Q(k+1))); (continued fraction). - Sergei N. Gladkovskii, Sep 14 2013

D-finite with recurrence: n*a(n) - 4*(n-1)*a(n-2) - 2*(2*n-3)*a(n-3)=0. - R. J. Mathar, Jan 14 2020