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A136029 a(n) is the number of central ideals of a garland of order 2n, i.e., a(n) = g(2n,n), where g(n,k) is the number of ideals of size k in a garland (or double fence) of order n (see A137278).

%I A136029 #36 Oct 18 2023 11:49:45
%S A136029 1,1,1,3,7,15,33,75,171,391,899,2077,4815,11195,26097,60975,142751,
%T A136029 334791,786419,1849905,4357121,10274313,24252923,57305241,135521807,
%U A136029 320758587,759757139,1800838381,4271267043,10136815015,24070870545
%N A136029 a(n) is the number of central ideals of a garland of order 2n, i.e., a(n) = g(2n,n), where g(n,k) is the number of ideals of size k in a garland (or double fence) of order n (see A137278).
%C A136029 Also the number of one-sided n-step prudent walks, starting from (0,0) and ending on the y-axis, with east, west and north steps. - _Shanzhen Gao_, Apr 26 2011
%D A136029 T. S. Blyth, J. C. Varlet, Ockham algebras, Oxford Science Pub. 1994.
%D A136029 E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.
%H A136029 Emanuele Munarini, <a href="/A136029/b136029.txt">Table of n, a(n) for n = 0..100</a>
%H A136029 Aubrey Blecher and Arnold Knopfmacher, <a href="https://doi.org/10.1007/s00010-023-01002-8">Prefixes of bargraph paths</a>, Aequationes Math. (2023).
%H A136029 Shanzhen Gao and Keh-Hsun Chen, <a href="http://worldcomp-proceedings.com/proc/p2014/FCS2696.pdf">Tackling Sequences From Prudent Self-Avoiding Walks</a>, FCS'14, The 2014 International Conference on Foundations of Computer Science.
%H A136029 S. Gao and H. Niederhausen, <a href="http://math.fau.edu/Niederhausen/HTML/Papers/Sequences%20Arising%20From%20Prudent%20Self-Avoiding%20Walks-February%2001-2010.pdf">Sequences Arising From Prudent Self-Avoiding Walks</a>, 2010.
%H A136029 Emanuele Munarini, <a href="http://www.emis.de/journals/INTEGERS/papers/j29/j29.Abstract.html">Combinatorial properties of the antichains of a garland</a>, Integers, 9 (2009), 353-374.
%F A136029 Recurrence: (n+6)*a(n+6) - (2*n+11)*a(n+5) - (n+3)*a(n+4) - 4*a(n+3) - (n+4)*c_(n+2) - (2*n+3)*a(n+1) + (n+1), a(n) = 0.
%F A136029 G.f.: (1 - x^2)/sqrt( 1 - 2*x - x^2 - x^4 + 2*x^5 + x^6 ).
%F A136029 a(n) = 1+sum(k=1..floor((n-1)/2), sum(i=1..min(n-2*k,k), C(n-2*k+1,i) * C(k-1,k-i) * C(n-k-i,k) ) ). - _Shanzhen Gao_, May 13 2011
%e A136029 a(4) = 7, since the central ideals of the garland G(4):
%e A136029 5..6..7..8
%e A136029 o..o..o..o
%e A136029 |\/|\/|\/|
%e A136029 |/\|/\|/\|
%e A136029 o..o..o..o
%e A136029 1..2..3..4
%e A136029 are: 1234, 1253, 1254, 1236, 2347, 1348, 2348.
%e A136029 a(4)=7, since there are 7 such walks: NNNN, NENW, NWNE, ENWN, ENNW, WNEN, WNNE. - _Shanzhen Gao_, May 13 2011
%K A136029 easy,nonn
%O A136029 0,4
%A A136029 _Emanuele Munarini_, Mar 21 2008