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A114495
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Number of returns to the x-axis in all hill-free Dyck paths of semilength n (a Dyck path is said to be hill-free if it has no peaks at level 1).
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2
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0, 1, 2, 7, 22, 73, 246, 844, 2936, 10334, 36736, 131709, 475714, 1729345, 6322534, 23232616, 85757008, 317839438, 1182341740, 4412949358, 16521076012, 62024023306, 233451103612, 880764587512, 3330234867792, 12617475113968
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,3
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COMMENTS
| Row sums of A114494.
Self-convolution of A000958 [From Sergio Falcon (sfalcon(AT)dma.ulpgc.es), Oct 28 2008]
Removing the initial zeros and setting both offsets to zero, this here is the Catalan transform of A006918. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jun 29 2009]
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FORMULA
| a(n)=sum(k^2*binomial(2n-2k, n-2k)/(n-k), k=1..floor(n/2)). G.f.=[1-sqrt(1-4z)]^2/[1+sqrt(1-4z)+2z]^2.
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EXAMPLE
| a(4)=7 because in the six hill-free Dyck paths of semilength 4, namely
UUD(D)UUD(D), UUDUDUD(D), UUDUUDD(D), UUUDDUD(D), UUUDUDD(D) and UUUUDDD(D), we have alltogether 7 returns to the x-axis (shown between parentheses).
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MAPLE
| a:=n->sum(k^2*binomial(2*n-2*k, n-2*k)/(n-k), k=1..floor(n/2)): seq(a(n), n=1..30);
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CROSSREFS
| Cf. A114494.
Sequence in context: A030186 A162770 A116387 * A137398 A151439 A204218
Adjacent sequences: A114492 A114493 A114494 * A114496 A114497 A114498
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Dec 01 2005
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