A227081 Row sums of A124576.

1, 2, 8, 40, 212, 1152, 6360, 35520, 200132, 1135456, 6478088, 37128896, 213617704, 1233014720, 7136819376, 41408161920, 240758343684, 1402436532576, 8182797500328, 47814708577728, 279768031296312, 1638915078384960, 9611453035886160

Offset: 0

Author

R. J. Mathar, Jun 30 2013

Keywords

• nonn

Comments

The offset is chosen following the Deleham offset in A124576, not according to the less systematic offset in the definition.

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• Isaac DeJager, Madeleine Naquin, Frank Seidl, Colored Motzkin Paths of Higher Order, VERUM 2019.

Programs

Magma

m:=30; R:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/(6*x -1 +2*Sqrt((2*x-1)*(6*x-1))) )); // G. C. Greubel, Nov 19 2018

Maple

AA := proc(n,k,x,y)
    option remember;
    if k <0 or k > n then
        0 ;
    elif n = 0 then
        1;
    elif k = 0 then
        x*procname(n-1,k,x,y)+procname(n-1,1,x,y) ;
    else
        procname(n-1,k-1,x,y)+y*procname(n-1,k,x,y)+procname(n-1,k+1,x,y) ;
    end if;
end proc:
seq(add( AA(n,k,1,4),k=0..n),n=0..30) ;

Mathematica

CoefficientList[Series[1/(6*x-1+2*Sqrt[(2*x-1)*(6*x-1)]), {x, 0, 30}], x] (* Vaclav Kotesovec, Jul 06 2013 *)

PARI

x='x+O('x^30); Vec(1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1)))) \\ G. C. Greubel, Nov 19 2018

Sage

s= (1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1)))).series(x,30); s.coefficients(x, sparse=False) # G. C. Greubel, Nov 19 2018

Formula

Conjecture: 3*n*a(n) +2*(-13*n+9)*a(n-1) +4*(13*n-21)*a(n-2) +24*(-n+2)*a(n-3)=0.

a(n) ~ 2^(n-3/2)*3^(n+1/2)/sqrt(Pi*n). - Vaclav Kotesovec, Jul 06 2013

G.f.: 1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1))). - Vaclav Kotesovec, Jul 06 2013