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A075125
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Number of staircase polyominoes of site-perimeter n.
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1
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0, 0, 0, 1, 0, 2, 2, 5, 10, 21, 46, 102, 230, 526, 1216, 2838, 6678, 15825, 37734, 90469, 217962, 527418, 1281250, 3123603, 7639784, 18740795, 46096732, 113666820, 280928470, 695796891
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,6
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COMMENTS
| a(n) = number of Dyck n-paths with no UDUs and no DUDs (A004148) whose first ascent is of length 3. For example, a(5)=2 counts UUUDDUUDDD, UUUDDDUUDD. - David Callan (callan(AT)stat.wisc.edu), May 08 2007
Contribution from Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009: (Start)
a(n)=Sum(k*A166299(n-2,k), k>=0).
Number of UUDD's starting at level 0 in all Dyck paths of semilength n-2 that have no ascents and no descents of length 1. Example: a(6)=2 because in UUDDUUDD and UUUUDDDD we have 2 + 0 = 2 UUDD's starting at level 0. (The Dyck paths having no ascents and no descents of length 1 are enumerated by the secondary structure numbers A004148).
(End)
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REFERENCES
| M. P. Delest, D. Gouyou-Beauchamps and B. Vauquelin, Enumeration of parallelogram polyominoes with given bond and site parameter, Graphs and Combinatorics, 3(1987),325-339. [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009]
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LINKS
| M. Bousquet-Melou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
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FORMULA
| G.F.: p^2/2*(1-p^2-2*p^3+p^4-(1+p-p^2)*sqrt((1+p+p^2)*(1-3*p+p^2)));
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MAPLE
| G := 4*z^4/(1+z-z^2+sqrt((1+z+z^2)*(1-3*z+z^2)))^2: Gser := series(G, z = 0, 32): seq(coeff(Gser, z, n), n = 1 .. 30); [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009]
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CROSSREFS
| Cf. A004148, A166299 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009]
Sequence in context: A184713 A110182 A193899 * A081374 A117400 A005637
Adjacent sequences: A075122 A075123 A075124 * A075126 A075127 A075128
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KEYWORD
| nonn
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AUTHOR
| Andrew Rechnitzer (a.rechnitzer(AT)ms.unimelb.edu.au), Sep 09 2002
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EXTENSIONS
| Offset changed to 1 by Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009
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