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Search: id:A075125
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| A075125 |
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Number of staircase polyominoes of site-perimeter n. |
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+0
2
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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; listen)
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OFFSET
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1,6
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COMMENT
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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
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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
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M. Bousquet-Melou and A. Rechnitzer, The site-perimeter of bargraphs, Adv. in Appl. Math. 31 (2003), 86-112.
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FORMULA
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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
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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
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Cf. A004148, A166299 [From Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009]
Sequence in context: A060405 A003228 A110182 this_sequence A081374 A117400 A005637
Adjacent sequences: A075122 A075123 A075124 this_sequence A075126 A075127 A075128
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KEYWORD
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nonn,new
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AUTHOR
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Andrew Rechnitzer (a.rechnitzer(AT)ms.unimelb.edu.au), Sep 09 2002
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EXTENSIONS
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Offset changed to 1 by Emeric Deutsch (deutsch(AT)duke.poly.edu), Nov 07 2009
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