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A129167
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Number of base pyramids in all skew Dyck paths of semilength n. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis.
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1
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0, 1, 3, 9, 30, 109, 420, 1685, 6960, 29391, 126291, 550359, 2426502, 10803801, 48507843, 219377949, 998436792, 4569488371, 21016589073, 97090411019, 450314942682, 2096122733211, 9788916220518, 45850711498859
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=Sum(k*A129165(n,k),k=0..n). a(n)=absolute value of A091699(n+1). Partial sums of A033321(n), n=1,2,3,...
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LINKS
| E. Deutsch, E. Munarini, S. Rinaldi, Skew Dyck paths, J. Stat. Plann. Infer. 140 (8) (2010) 2191-2203
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FORMULA
| G.f.=[1-3z-sqrt(1-6z+5z^2)]/[z(3-3z-sqrt(1-6z+5z^2))].
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EXAMPLE
| a(2)=3 because in the paths (UD)(UD), (UUDD) and UUDL we have altogether 3 base pyramids (shown between parentheses).
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MAPLE
| G:=(1-3*z-sqrt(1-6*z+5*z^2))/z/(3-3*z-sqrt(1-6*z+5*z^2)): Gser:=series(G, z=0, 30): seq(coeff(Gser, z, n), n=0..27);
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CROSSREFS
| Cf. A129165, A091699, A033321.
Sequence in context: A200074 A032125 A091699 * A151472 A107379 A117428
Adjacent sequences: A129164 A129165 A129166 * A129168 A129169 A129170
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Apr 04 2007
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