Search: id:A054341
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%I A054341
%S A054341 1,2,5,12,30,74,185,460,1150,2868,7170,17904,44760,111834,279585,
%T A054341 698748,1746870,4366460,10916150,27287944,68219860,170541252,426353130,
%U A054341 1065853432,2664633580,6661479944,16653699860,41633878200,104084695500
%N A054341 Row sums of triangle A054336 (central binomial convolutions).
%C A054341 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 (callan(AT)stat.wisc.edu), Mar 02 2005
%C A054341 Inverse Chebyshev transform of the second kind applied to 2^n. This maps 
               g(x)->c(x^2)g(xc(x^2)). - Paul Barry (pbarry(AT)wit.ie), Sep 14 2005
%C A054341 Hankel transform of this sequence gives A000012 = [1,1,1,1,1,1,1,...] 
               . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Oct 24 2007
%C A054341 Inverse binomial transform of A059738. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 24 2009]
%H A054341 J. W. Layman, <a href="http://www.cs.uwaterloo.ca/journals/JIS/index.html">
               The Hankel Transform and Some of its Properties</a>, J. Integer Sequences, 
               4 (2001), #01.1.5.
%F A054341 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.
%F A054341 G.f.: c(x^2)/(1-2xc(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 (pbarry(AT)wit.ie), Sep 14 2005
%F A054341 a(n)=A127358(n+1)-2*A127358(n). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Mar 02 2007
%F A054341 a(n)=A126075(n,0). [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 
               24 2009]
%F A054341 a(n)= Sum_{k, 0<=k<=n} A053121(n,k)*2^k. [From Philippe DELEHAM (kolotoko(AT)wanadoo.fr), 
               Nov 28 2009]
%Y A054341 Cf. A000108, A054336.
%Y A054341 Sequence in context: A062423 A118649 A033482 this_sequence A000106 A076883 
               A140832
%Y A054341 Adjacent sequences: A054338 A054339 A054340 this_sequence A054342 A054343 
               A054344
%K A054341 easy,nonn,new
%O A054341 0,2
%A A054341 Wolfdieter Lang (wolfdieter.lang(AT)physik.uni-karlsruhe.de), Mar 13 
               2000

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