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A076026
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G.f.: (1-4*x*C)/(1-5*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108.
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7
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1, 1, 6, 37, 230, 1434, 8952, 55917, 349374, 2183230, 13643972, 85270626, 532926716, 3330739972, 20816939100, 130105200765, 813155081070, 5082210417270, 31763782696740, 198523522444950, 1240771573465140, 7754820693127020, 48467623215477120, 302922622226091090
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n) is the number of Motzkin paths of length n-1 in which the (1,0)-steps at level 0 come in 6 colors and those at a higher level come in 2 colors. Example: a(4)=230 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 6^3 = 216 paths of shape HHH, 6 paths of shape HUD, 6 paths of shape UDH, and 2 paths of shape UHD. - Emeric Deutsch, May 02 2011
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REFERENCES
| L. W. Shapiro and C. J. Wang, Generating identities via 2 X 2 matrices, Congressus Numerantium, 205 (2010), 33-46.
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FORMULA
| a(n+1)=Sum_{k, 0<=k<=n}A039598(n,k)*4^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Mar 21 2007
a(n) = Sum_{k, 0<=k<=n}A039599(n,k)*A015521(k), for n>=1 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 22 2007
Let A be the Toeplitz matrix of order n defined by: A[i,i-1]=-1, A[i,j]=Catalan(j-i), (i<=j), and A[i,j]=0, otherwise. Then, for n>=1, a(n+1)=(-1)^n*charpoly(A,-5). [From Milan R. Janjic (agnus(AT)blic.net), Jul 08 2010]
From Gary W. Adamson, Jul 25 2011: (start) a(n) = upper left term in M^(n-1), M = an infinite square production matrix as follows:
6, 1, 0, 0, 0,...
1, 1, 1, 0, 0,...
1, 1, 1, 1, 0,...
1, 1, 1, 1, 1,...
... (end)
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CROSSREFS
| Cf. A000108, A001700, A049027, A076025.
Sequence in context: A005668 A018904 A192807 * A161734 A081570 A122898
Adjacent sequences: A076023 A076024 A076025 * A076027 A076028 A076029
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com), Oct 29 2002
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