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 A137278 Triangle read by rows: g(n,k) = number of ideals of size k in a garland (or double fence) of order n. 3
 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 3, 3, 3, 1, 1, 4, 6, 6, 7, 6, 6, 4, 1, 1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1, 1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1, 1, 7, 21, 37, 50, 63, 72, 75, 72, 63, 50, 37, 21, 7, 1, 1, 8, 28, 58, 88, 118, 146, 164, 171, 164, 146, 118, 88, 58, 28, 8, 1 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,6 COMMENTS Row n has 2n+1 terms. Also triangle of bounded variation linear paths of length n having final height k-n (height varies from -n to n). Olivier Gérard, Aug 28 2012 Bounded variation linear paths are path formed from steps 0,1,-1 where the step successions (-1,1) or (1,-1) are not allowed. Equivalently ternary strings of length n with subwords (0,2) and (2,0) not allowed and total sum k. REFERENCES T. S. Blyth and J. C. Varlet, Ockham Algebras, Oxford Science Pub. 1994. E. Munarini, Enumeration of order ideals of a garland, Ars Combin. 76 (2005), 185-192. LINKS _Emanuele Munarini_, Mar 13 2008, Table of n, a(n) for n = 0..440 [Rows 0 through 20, flattened] FORMULA G.f.: G(x,t) = (1-x^2*t^2)/(1-(1+x+x^2)*t+x^2*t^2+x^3*t^3). Recurrence: g(n+3,k+3) = g(n+2,k+3) + g(n+2,k+2) + g(n+2,k+1) - g(n+1,k+1) - g(n,k) EXAMPLE In the garland 5..6..7..8 o..o..o..o |\/|\/|\/| |/\|/\|/\| o..o..o..o 1..2..3..4 the ideals of size 4 are 1234, 1253, 1254, 1236, 2347, 1348, 2348. The ternary strings of size 4 with total sum 4 are 0022, 0202, 0220, 2002, 2020, 2200, 0112, 0121, 0211, 1012, 1021, 2011, 1102, 1201, 2101, 1120, 1210, 2110, 1111 Applying the restriction gives 7 possible strings 0112, 0121, 1012, 2101, 1210, 2110, 1111 Triangle begins: 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 3, 3, 3, 3, 1, 1, 4, 6, 6, 7, 6, 6, 4, 1, 1, 5, 10, 12, 14, 15, 14, 12, 10, 5, 1, 1, 6, 15, 22, 27, 32, 33, 32, 27, 22, 15, 6, 1, ... CROSSREFS Sequence of row sums is A001333 / A078057. Sequence in context: A059780 A075119 A224076 * A205810 A139368 A134303 Adjacent sequences:  A137275 A137276 A137277 * A137279 A137280 A137281 KEYWORD easy,nonn,tabf AUTHOR Emanuele Munarini, Mar 13 2008 STATUS approved

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