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%I #58 Jan 23 2024 09:18:38
%S 1,2,5,12,30,74,185,460,1150,2868,7170,17904,44760,111834,279585,
%T 698748,1746870,4366460,10916150,27287944,68219860,170541252,
%U 426353130,1065853432,2664633580,6661479944,16653699860,41633878200,104084695500,260210401530,650526003825
%N Row sums of triangle A054336 (central binomial convolutions).
%C 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
%C 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
%C 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
%C Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...]. - _Philippe Deléham_, Oct 24 2007
%C Inverse binomial transform of A059738. - _Philippe Deléham_, Nov 24 2009
%H Alois P. Heinz, <a href="/A054341/b054341.txt">Table of n, a(n) for n = 0..2513</a> (first 201 terms from Vincenzo Librandi)
%H Paul Barry, <a href="https://www.emis.de/journals/JIS/VOL22/Barry1/barry411.html">The Central Coefficients of a Family of Pascal-like Triangles and Colored Lattice Paths</a>, J. Int. Seq., Vol. 22 (2019), Article 19.1.3.
%H Isaac DeJager, Madeleine Naquin, Frank Seidl, <a href="https://www.valpo.edu/mathematics-statistics/files/2019/08/Drube2019.pdf">Colored Motzkin Paths of Higher Order</a>, VERUM 2019.
%H J. W. Layman, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL4/LAYMAN/hankel.html">The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 4 (2001), #01.1.5.
%F a(n) = Sum_{m=0..n} A054336(n, m).
%F G.f.: 1/(1-2*x-x^2*c(x^2)), where c(x) = g.f. for Catalan numbers A000108.
%F From_Paul Barry_, Sep 14 2005: (Start)
%F G.f.: c(x^2)/(1-2*x*c(x^2));
%F a(n) = Sum_{k=0..n} binomial(n,(n-k)/2)*(1 + (-1)^(n+k))*2^k*(k+1)/(n+k+2). (End)
%F G.f.: 2/(1-4*x+sqrt(1-4*x^2)). - _Ira M. Gessel_, Oct 27 2013
%F a(n) = A127358(n+1) - 2*A127358(n). - _Philippe Deléham_, Mar 02 2007
%F a(n) = A126075(n,0). - _Philippe Deléham_, Nov 24 2009
%F a(n) = Sum_{k=0..n} A053121(n,k)*2^k. - _Philippe Deléham_, Nov 28 2009
%F From _Gary W. Adamson_, Sep 07 2011: (Start)
%F a(n) is the upper left term of M^n, M is an infinite square production matrix as follows:
%F 2, 1, 0, 0, 0, ...
%F 1, 0, 2, 0, 0, ...
%F 0, 1, 0, 1, 0, ...
%F 0, 0, 1, 0, 1, ...
%F 0, 0, 0, 1, 0, ...
%F ... (End)
%F 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
%F a(n) ~ 3 * 5^n / 2^(n+2). - _Vaclav Kotesovec_, Feb 13 2014
%e a(4) = 30, the upper left term of M^4.
%p b:= proc(x, y) option remember; `if`(x=0, 1,
%p b(x-1, 0)+`if`(y>0, b(x-1, y-1), 0)+b(x-1, y+1))
%p end:
%p a:= n-> b(n, 0):
%p seq(a(n), n=0..31); # _Alois P. Heinz_, Jan 23 2024
%t CoefficientList[Series[2/(1-4*x+Sqrt[1-4*x^2]), {x, 0, 20}], x] (* _Vaclav Kotesovec_, Feb 13 2014 *)
%Y Cf. A000108, A054336.
%K easy,nonn
%O 0,2
%A _Wolfdieter Lang_, Mar 13 2000
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