A033543 Expansion of (1 - sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)).

1, 2, 5, 16, 62, 270, 1257, 6096, 30398, 154756, 800834, 4199720, 22269976, 119207942, 643277553, 3495713184, 19113486390, 105074982876, 580435709622, 3220217022144, 17935186513044, 100243540330188, 562080274898250, 3160904659483104, 17823384503589996, 100749266778698280

Offset: 0

Author

N. J. A. Sloane

Keywords

• nonn

Comments

Binomial transform of A033321. - Philippe Deléham, Nov 26 2009

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

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.
• J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.

Programs

Magma

R:=PowerSeriesRing(Rationals(), 40); Coefficients(R!( (1-Sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) )); // G. C. Greubel, Oct 12 2019

Maple

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

Mathematica

CoefficientList[Series[(1-Sqrt[(1-2x)(1-6x)])/(2x(2-3x)),{x,0,40}],x] (* Harvey P. Dale, Aug 12 2012 *)

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

Sage

def A033543_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ).list()
A033543_list(40) # G. C. Greubel, Oct 12 2019

Formula

a(n) = A124575(n,0). - Philippe Deléham, Nov 26 2009

a(n) = Sum_{k=0..n} A052179(n,k)*(-2)^k. - Philippe Deléham, Nov 28 2009

From Gary W. Adamson, Jul 21 2011: (Start)

a(n) = upper left term in M^n, M = an infinite square production matrix as follows (with the main diagonal (2,3,3,3,...)):

2, 1, 0, 0, ...

1, 3, 1, 0, ...

1, 1, 3, 1, ...

1, 1, 1, 3, ...

... (End)

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

a(n) ~ 6^(n+1/2)/(3*sqrt(Pi)*n^(3/2)). - Vaclav Kotesovec, Oct 08 2012