%I #20 Sep 08 2022 08:46:05

%S 1,2,8,40,212,1152,6360,35520,200132,1135456,6478088,37128896,

%T 213617704,1233014720,7136819376,41408161920,240758343684,

%U 1402436532576,8182797500328,47814708577728,279768031296312,1638915078384960,9611453035886160

%N Row sums of A124576.

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

%H Vincenzo Librandi, <a href="/A227081/b227081.txt">Table of n, a(n) for n = 0..200</a>

%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.

%F 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.

%F a(n) ~ 2^(n-3/2)*3^(n+1/2)/sqrt(Pi*n). - _Vaclav Kotesovec_, Jul 06 2013

%F G.f.: 1/(6*x -1 +2*sqrt((2*x-1)*(6*x-1))). - _Vaclav Kotesovec_, Jul 06 2013

%p AA := proc(n,k,x,y)

%p option remember;

%p if k <0 or k > n then

%p 0 ;

%p elif n = 0 then

%p 1;

%p elif k = 0 then

%p x*procname(n-1,k,x,y)+procname(n-1,1,x,y) ;

%p else

%p procname(n-1,k-1,x,y)+y*procname(n-1,k,x,y)+procname(n-1,k+1,x,y) ;

%p end if;

%p end proc:

%p seq(add( AA(n,k,1,4),k=0..n),n=0..30) ;

%t CoefficientList[Series[1/(6*x-1+2*Sqrt[(2*x-1)*(6*x-1)]), {x, 0, 30}], x] (* _Vaclav Kotesovec_, Jul 06 2013 *)

%o (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

%o (Magma) m:=30; R<x>:=PowerSeriesRing(Rationals(), m); Coefficients(R!( 1/(6*x -1 +2*Sqrt((2*x-1)*(6*x-1))) )); // _G. C. Greubel_, Nov 19 2018

%o (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

%K nonn

%O 0,2

%A _R. J. Mathar_, Jun 30 2013