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A127540
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Number of odd-length branches starting at the root in all ordered trees with n edges.
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3
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0, 1, 2, 7, 21, 69, 228, 773, 2659, 9275, 32715, 116511, 418377, 1513163, 5507242, 20155583, 74131537, 273862373, 1015762117, 3781095113, 14121051487, 52895245133, 198681804877, 748162728797, 2823879525331, 10681527145369
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| a(n)=Sum(k*A127538(n,k),k=0..n).
Also number of even-length branches starting at the root in all ordered trees with n+1 edges. - Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 03 2007
Also number of Dyck paths of semi-length n+1 with first descent and last ascent of equal size.
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FORMULA
| G.f.=zC/(1-z^2-zC-z^2*C), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
| a(2)=2 because the tree /\ has two odd-length branches starting from the root and the path-tree of length 2 has none.
a(2)=2 because the Dyck paths of semi-length 3 with first descent and last ascent of same size are UUDUDD and UDUDUD.
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MAPLE
| C:=(1-sqrt(1-4*z))/2/z: g:=z*C/(1-z^2-z*C-z^2*C): gser:=series(g, z=0, 32): seq(coeff(gser, z, n), n=0..29);
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CROSSREFS
| Cf. A127538.
Sequence in context: A052911 A126133 A186240 * A060900 A151289 A150300
Adjacent sequences: A127537 A127538 A127539 * A127541 A127542 A127543
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Mar 01 2007
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