%I #22 Sep 08 2022 08:45:29

%S 1,3,10,33,110,366,1220,4065,13550,45162,150540,501786,1672620,

%T 5575356,18584520,61948257,206494190,688313490,2294378300,7647926046,

%U 25493086820,84976950468,283256501560,944188318938,3147294396460

%N a(n) = A127359(n+1)/2 - A127359(n).

%C Hankel transform is A000012=[1,1,1,1,1,1,1,...].

%C a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and there are no (1,0)-steps at a higher level. Example: a(3)=33 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH, 3 paths of shape HUD, and 3 paths of shape UDH. - _Emeric Deutsch_, May 02 2011

%H Michael De Vlieger, <a href="/A126931/b126931.txt">Table of n, a(n) for n = 0..1912</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 G.f.: 1/(1-3x-x^2/(1-x^2/(1-x^2/(1-x^2/(1-... (continued fraction). - _Paul Barry_, Mar 10 2009

%F G.f.: 2/(1 - 6*x + sqrt(1-4*x^2)). - _Emeric Deutsch_, May 02 2011

%F Conjecture: 3*(n+1)*a(n) +10*(-n-1)*a(n-1) +12*(-n+2)*a(n-2) +40*(n-2)*a(n-3)=0. - _R. J. Mathar_, Nov 26 2012

%p A127359 := proc(n) add(binomial(n,floor(k/2))*3^(n-k), k=0..n) ; end proc: A126931 := proc(n) A127359(n+1)/2-A127359(n) ; end proc: seq(A126931(n),n=0..50) ; # _R. J. Mathar_, Mar 25 2010

%t With[{s = Partition[#, 2, 1] &@ Array[Sum[Binomial[#, Floor[k/2]]*3^(# - k), {k, 0, #}] &, 26, 0]}, Map[#2/2 - #1 & @@ # &, s]] (* _Michael De Vlieger_, Dec 15 2019 *)

%t CoefficientList[Series[2/(1-6*x+Sqrt[1-4*x^2]), {x,0,30}], x] (* _G. C. Greubel_, Jan 29 2020 *)

%o (PARI) my(x='x+O('x^30)); Vec( 2/(1 - 6*x + sqrt(1-4*x^2)) ) \\ _G. C. Greubel_, Jan 29 2020

%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 30); Coefficients(R!( 2/(1 - 6*x + Sqrt(1-4*x^2)) )); // _G. C. Greubel_, Jan 29 2020

%o (Sage)

%o def A126931_list(prec):

%o P.<x> = PowerSeriesRing(ZZ, prec)

%o return P( 2/(1 - 6*x + sqrt(1-4*x^2)) ).list()

%o A126931_list(30) # _G. C. Greubel_, Jan 29 2020

%Y Cf. A127359.

%K nonn

%O 0,2

%A _Philippe Deléham_, Mar 17 2007

%E More terms from _R. J. Mathar_, Mar 25 2010