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Search: id:A114515
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| A114515 |
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Number of peaks in all hill-free Dyck paths of semilength n (a Dyck path is hill-free if it has no peaks at level 1). |
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3
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| 0, 0, 1, 3, 12, 45, 171, 651, 2488, 9540, 36690, 141482, 546864, 2118207, 8219967, 31952115, 124389552, 484908408, 1892657934, 7395597354, 28928182440, 113260606074, 443827115886, 1740592240638, 6831289801872, 26829201570600
(list; graph; listen)
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OFFSET
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0,4
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COMMENT
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a(n)=Sum(k*A100754(n,k), k=0..n-1).
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FORMULA
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G.f.=z^2*C/[(1-zC+z)^2*(1-2zC)}, where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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a(3)=3 because in the two hill-free Dyck paths of semilength 3, namely U(UD)(UD)D and UU(UD)DD, we have alltogether 3 peaks (shown between parantheses).
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=1/(1-z*C+z)^2*z^2*C/(1-2*z*C): Gser:=series(G, z=0, 32): 0, seq(coeff(Gser, z^n), n=1..28);
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CROSSREFS
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Cf. A100754.
Adjacent sequences: A114512 A114513 A114514 this_sequence A114516 A114517 A114518
Sequence in context: A128593 A085481 A030195 this_sequence A151162 A094547 A026559
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KEYWORD
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nonn
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 04 2005
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