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A054341
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Row sums of triangle A054336 (central binomial convolutions).
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13
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1, 2, 5, 12, 30, 74, 185, 460, 1150, 2868, 7170, 17904, 44760, 111834, 279585, 698748, 1746870, 4366460, 10916150, 27287944, 68219860, 170541252, 426353130, 1065853432, 2664633580, 6661479944, 16653699860, 41633878200, 104084695500, 260210401530, 650526003825
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OFFSET
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0,2
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COMMENTS
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a(n) = # Dyck (n+1)-paths all of whose components are symmetric. A strict Dyck path is one with exactly one return to ground level (necessarily at the end). Every nonempty Dyck path is expressible uniquely as a concatenation of one or more strict Dyck paths, called its components. - David Callan, Mar 02 2005
a(n) = # 2-Motzkin paths (i.e., Motzkin paths with blue and red level steps) with no level steps at positive height. Example: a(2)=5 because, denoting U=(1,1), D=(1,-1), B=blue (1,0), R=red (1,0), we have BB, BR, RB, RR, and UD. - Emeric Deutsch, Jun 07 2011
Inverse Chebyshev transform of the second kind applied to 2^n. This maps g(x) -> c(x^2)g(xc(x^2)). - Paul Barry, Sep 14 2005
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LINKS
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FORMULA
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G.f.: 1/(1-2*x-x^2*c(x^2)), where c(x) = g.f. for Catalan numbers A000108.
From_Paul Barry_, Sep 14 2005: (Start)
G.f.: c(x^2)/(1-2*x*c(x^2));
a(n) = Sum_{k=0..n} binomial(n,(n-k)/2)*(1 + (-1)^(n+k))*2^k*(k+1)/(n+k+2). (End)
a(n) is the upper left term of M^n, M is an infinite square production matrix as follows:
2, 1, 0, 0, 0, ...
1, 0, 2, 0, 0, ...
0, 1, 0, 1, 0, ...
0, 0, 1, 0, 1, ...
0, 0, 0, 1, 0, ...
... (End)
Conjecture: 2*(n+1)*a(n) +5*(-n-1)*a(n-1) +8*(-n+2)*a(n-2) +20*(n-2)*a(n-3)=0. - R. J. Mathar, Nov 30 2012
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EXAMPLE
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a(4) = 30, the upper left term of M^4.
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MAPLE
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b:= proc(x, y) option remember; `if`(x=0, 1,
b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
end:
a:= n-> b(n, 0):
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MATHEMATICA
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CoefficientList[Series[2/(1-4*x+Sqrt[1-4*x^2]), {x, 0, 20}], x] (* Vaclav Kotesovec, Feb 13 2014 *)
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CROSSREFS
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KEYWORD
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easy,nonn
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AUTHOR
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STATUS
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approved
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