A137278 -id:A137278 - cp's OEIS frontend

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).

1, 1, 1, 3, 7, 15, 33, 75, 171, 391, 899, 2077, 4815, 11195, 26097, 60975, 142751, 334791, 786419, 1849905, 4357121, 10274313, 24252923, 57305241, 135521807, 320758587, 759757139, 1800838381, 4271267043, 10136815015, 24070870545

Offset: 0

Emanuele Munarini, Mar 21 2008

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

			a(4) = 7, since the central ideals of the garland G(4):
5..6..7..8
o..o..o..o
|\/|\/|\/|
|/\|/\|/\|
o..o..o..o
1..2..3..4
are: 1234, 1253, 1254, 1236, 2347, 1348, 2348.
a(4)=7, since there are 7 such walks: NNNN, NENW, NWNE, ENWN, ENNW, WNEN, WNNE. - _Shanzhen Gao_, May 13 2011

T. S. Blyth, J. C. Varlet, Ockham algebras, Oxford Science Pub. 1994.
E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192.

Emanuele Munarini, Table of n, a(n) for n = 0..100
Aubrey Blecher and Arnold Knopfmacher, Prefixes of bargraph paths, Aequationes Math. (2023).
Shanzhen Gao and Keh-Hsun Chen, Tackling Sequences From Prudent Self-Avoiding Walks, FCS'14, The 2014 International Conference on Foundations of Computer Science.
S. Gao and H. Niederhausen, Sequences Arising From Prudent Self-Avoiding Walks, 2010.
Emanuele Munarini, Combinatorial properties of the antichains of a garland, Integers, 9 (2009), 353-374.

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.
G.f.: (1 - x^2)/sqrt( 1 - 2*x - x^2 - x^4 + 2*x^5 + x^6 ).
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

A136018 Triangle read by rows: r(n,k) = g(n,n-k), where g(n,k) is the number of ideals of size k in a garland (or double fence) of order n (see A137278).

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 3, 3, 3, 1, 7, 6, 6, 4, 1, 15, 14, 12, 10, 5, 1, 33, 32, 27, 22, 15, 6, 1, 75, 72, 63, 50, 37, 21, 7, 1, 171, 164, 146, 118, 88, 58, 28, 8, 1, 391, 377, 338, 280, 212, 147, 86, 36, 9, 1, 899, 870, 786, 662, 514, 366, 234, 122, 45, 10, 1, 2077, 2014, 1834, 1564

Offset: 0

Emanuele Munarini, Mar 21 2008

Row n has n+1 terms.

T. S. Blyth, J. C. Varlet, Ockham algebras, Oxford Science Pub. 1994.
E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185--192.

Emanuele Munarini, Mar 21 2008, Table of n, a(n) for n = 0..495

Recurrence: r(n+3,k+1) = r(n+2,k) + r(n+2,k+1) + r(n+2,k+2) - r(n+1,k+1) - r(n,k+1).

Riordan matrix: R = ( g(x), f(x) ), where g(x) = ( 1 - x^2 )/sqrt( 1 - 2 x - x^2 - x^4 + 2 x^5 + x^6 ) f(x) = ( 1 - x + x^2 + x^3 - sqrt( 1 - 2 x - x^2 - 3 x^4 + 2 x^5 + x^6 ) )/(2x) g(x) is the generating series for the central ideals c(n) = g(2n,n). f(x)/x is the generating series for sequence A004149.

cp's OEIS Frontend

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).

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