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Search: id:A114590
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| A114590 |
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Number of peaks at even levels in all hill-free Dyck paths of semilength n+2 (a hill in a Dyck path is a peak at level 1). |
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3
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| 1, 2, 8, 28, 103, 382, 1432, 5408, 20546, 78436, 300636, 1156188, 4459267, 17241526, 66807856, 259361920, 1008598126, 3928120924, 15319329472, 59817190552, 233826979750, 914962032172, 3583556424208, 14047386554368, 55108441878868
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OFFSET
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0,2
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COMMENT
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a(n)=Sum(k*A114588(n+2,k),k=0..n+1).
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FORMULA
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G.f.=[1+2z^2-(1+2z)sqrt(1-4z)]/[2z^2*(2+z)^2*sqrt(1-4z)].
a(n)=sum{k=0..n, sum{j=0..n-k, C(n-j,k-j)*C(n-j,k)*(j+1)}}; - Paul Barry (pbarry(AT)wit.ie), Nov 03 2006
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EXAMPLE
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a(1)=2 because in the 2 (=A000957(4)) hill-free Dyck paths of semilength 3, namely UUUDDD and U(UD)(UD)D (U=(1,1), D=(1,-1)) we have altogether 2 peaks at even level (shown between parentheses).
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MAPLE
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G:=(1+2*z^2-(1+2*z)*sqrt(1-4*z))/2/z^2/(2+z)^2/sqrt(1-4*z): Gser:=series(G, z=0, 30): 1, seq(coeff(Gser, z^n), n=1..25);
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CROSSREFS
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Cf. A114588, A114587, A114515.
Adjacent sequences: A114587 A114588 A114589 this_sequence A114591 A114592 A114593
Sequence in context: A066796 A104934 A056711 this_sequence A133592 A115967 A122447
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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 11 2005
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