OFFSET
0,2
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.
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
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 *)
PROG
(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
(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
(Sage)
def A033543_list(prec):
P.<x> = 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
CROSSREFS
KEYWORD
nonn
AUTHOR
STATUS
approved