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A033543
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Expansion of (1 - sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)).
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7
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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
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OFFSET
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0,2
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COMMENTS
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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
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LINKS
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FORMULA
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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
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MAPLE
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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
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MATHEMATICA
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CoefficientList[Series[(1-Sqrt[(1-2x)(1-6x)])/(2x(2-3x)), {x, 0, 40}], x] (* Harvey P. Dale, Aug 12 2012 *)
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PROG
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(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)
P.<x> = PowerSeriesRing(ZZ, prec)
return P( (1-sqrt((1-2*x)*(1-6*x)))/(2*x*(2-3*x)) ).list()
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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