|
%I #44 Sep 08 2022 08:44:51
%S 1,2,5,16,62,270,1257,6096,30398,154756,800834,4199720,22269976,
%T 119207942,643277553,3495713184,19113486390,105074982876,580435709622,
%U 3220217022144,17935186513044,100243540330188,562080274898250,3160904659483104,17823384503589996,100749266778698280
%N Expansion of (1 - sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)).
%C Binomial transform of A033321. - _Philippe Deléham_, Nov 26 2009
%C a(n) is the number of Motzkin paths of length n in which the (1,0)-steps at level 0 come in 2 colors and those at a higher level come in 4 colors. Example: a(3)=16 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 2^3 = 8 paths of shape HHH, 2 paths of shape HUD, 2 paths of shape UDH, and 4 paths of shape UHD. - _Emeric Deutsch_, May 02 2011
%H Vincenzo Librandi, <a href="/A033543/b033543.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.
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F a(n) = A124575(n,0). - _Philippe Deléham_, Nov 26 2009
%F a(n) = Sum_{k=0..n} A052179(n,k)*(-2)^k. - _Philippe Deléham_, Nov 28 2009
%F From _Gary W. Adamson_, Jul 21 2011: (Start)
%F a(n) = upper left term in M^n, M = an infinite square production matrix as follows (with the main diagonal (2,3,3,3,...)):
%F 2, 1, 0, 0, ...
%F 1, 3, 1, 0, ...
%F 1, 1, 3, 1, ...
%F 1, 1, 1, 3, ...
%F ... (End)
%F D-finite with recurrence: 2*(n+1)*a(n) = (19*n-5)*a(n-1) - 12*(4*n-5)*a(n-2) + 36*(n-2)*a(n-3). - _Vaclav Kotesovec_, Oct 08 2012
%F a(n) ~ 6^(n+1/2)/(3*sqrt(Pi)*n^(3/2)). - _Vaclav Kotesovec_, Oct 08 2012
%p seq(coeff(series((1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)), x, n+2), x, n), n = 0..40); # _G. C. Greubel_, Oct 12 2019
%t CoefficientList[Series[(1-Sqrt[(1-2x)(1-6x)])/(2x(2-3x)),{x,0,40}],x] (* _Harvey P. Dale_, Aug 12 2012 *)
%o (PARI) x='x+O('x^66); Vec( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ) \\ _Joerg Arndt_, May 04 2013
%o (Magma) R<x>:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) )); // _G. C. Greubel_, Oct 12 2019
%o (Sage)
%o def A033543_list(prec):
%o P.<x> = PowerSeriesRing(ZZ, prec)
%o return P( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ).list()
%o A033543_list(40) # _G. C. Greubel_, Oct 12 2019
%K nonn
%O 0,2
%A _N. J. A. Sloane_
|
