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A054341
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Row sums of triangle A054336 (central binomial convolutions).
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9
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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
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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, June 7, 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
Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...] . - Philippe DELEHAM, Oct 24 2007
Inverse binomial transform of A059738. [From Philippe DELEHAM, Nov 24 2009]
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LINKS
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Table of n, a(n) for n=0..28.
J. W. Layman, The Hankel Transform and Some of its Properties, J. Integer Sequences, 4 (2001), #01.1.5.
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FORMULA
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a(n)= sum(A054336(n, m), m=0..n). G.f.: 1/(1-2*x-x^2*c(x^2)), where c(x) = g.f. for Catalan numbers A000108.
G.f.: c(x^2)/(1-2*x*c(x^2)); a(n)=sum{k=0..n, C(n, (n-k)/2)(1+(-1)^(n+k))2^k*(k+1)/(n+k+2)}. - Paul Barry, Sep 14 2005
a(n)=A127358(n+1)-2*A127358(n). - Philippe DELEHAM, Mar 02 2007
a(n)=A126075(n,0). [From Philippe DELEHAM, Nov 24 2009]
a(n)= Sum_{k, 0<=k<=n} A053121(n,k)*2^k. [From Philippe DELEHAM, Nov 28 2009]
a(n) is the upper left term of M^n, M = 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,...
... - Gary W. Adamson, Sep 07 2011
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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CROSSREFS
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Cf. A000108, A054336.
Sequence in context: A062423 A118649 A033482 * A000106 A076883 A140832
Adjacent sequences: A054338 A054339 A054340 * A054342 A054343 A054344
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KEYWORD
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easy,nonn
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AUTHOR
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Wolfdieter Lang, Mar 13 2000
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STATUS
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approved
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