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A114587
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Number of peaks at odd levels in all hill-free Dyck paths of semilength n+3 (a hill in a Dyck path is a peak at level 1).
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3
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1, 4, 17, 68, 269, 1056, 4132, 16144, 63046, 246228, 962019, 3760700, 14710589, 57581696, 225546488, 884059808, 3467476430, 13608852968, 53443415522, 210000136136, 825630208466, 3247733377664, 12781815016232, 50328168273408
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| a(n)=Sum(k*A114586(n+3,k),k=0..n+1).
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FORMULA
| G.f.=[1-2z-3z^2-2z^3-(1-z^2)sqrt(1-4z)]/[2z^4*(2+z)^2*sqrt(1-4z)].
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EXAMPLE
| a(1)=4 because in the 6 (=A000957(5)) hill-free Dyck paths of semilength 4, namely UUUUDDDD, UU(UD)(UD)DD, UUDU(UD)DD, UUDUDUDD, UU(UD)DUDD and UUDDUUDD (U=(1,1), D=(1,-1)) we have altogether 4 peaks at odd level (shown between parentheses).
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MAPLE
| G:=(1-2*z-3*z^2-2*z^3-(1-z^2)*sqrt(1-4*z))/2/sqrt(1-4*z)/z^4/(2+z)^2: Gser:=series(G, z=0, 32): 1, seq(coeff(Gser, z^n), n=1..26);
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CROSSREFS
| Cf. A114586, A114590, A114515.
Sequence in context: A046723 A030529 A081113 * A033114 A096881 A033122
Adjacent sequences: A114584 A114585 A114586 * A114588 A114589 A114590
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 11 2005
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