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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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6
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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
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| Binomial transform of A033321. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 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
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LINKS
| J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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FORMULA
| a(n)=A124575(n,0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2009]
a(n)= Sum_{k, 0<=k<=n} A052179(n,k)*(-2)^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 28 2009]
Contribution 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)
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CROSSREFS
| Sequence in context: A163747 A007976 A058259 * A124531 A129578 A005387
Adjacent sequences: A033540 A033541 A033542 * A033544 A033545 A033546
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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