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A033297
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Number of ordered rooted trees with n edges such that the rightmost leaf of each subtree is at even level. Equivalently, number of Dyck paths of semilength n with no return descents of odd length.
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5
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1, 1, 4, 10, 32, 100, 329, 1101, 3761, 13035, 45751, 162261, 580639, 2093801, 7601044, 27756626, 101888164, 375750536, 1391512654, 5172607766, 19293659254, 72188904386, 270870709264, 1019033438060, 3842912963392
(list; graph; refs; listen; history; internal format)
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OFFSET
| 2,3
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COMMENTS
| Prime p divides a(p-1) and a(p+1) for odd primes where 5 is a square mod p (A038872(k)). - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 01 2006
Hankel transform of 1,1,4,.. is A167477.
Hankel transform of a(n+1) (starts 0,1,1,4...) is -F(2n). [From Paul Barry (pbarry(AT)wit.ie), Dec 16 2008]
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| Sum((-1)^i*C(n-1-i), i=0..n-2), where C(n) are the Catalan numbers; g.f. = (1 - 2z - sqrt(1 - 4z))/(2(1+z); sums of two consecutive terms are the Catalan numbers )
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MATHEMATICA
| Table[Sum[(-1)^(n+k)*(2k)!/k!/(k+1)!, {k, 1, n}], {n, 1, 72}] - Alexander Adamchuk (alex(AT)kolmogorov.com), Jul 01 2006
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CROSSREFS
| Cf. A000108, A038872.
Sequence in context: A034717 A001673 A017936 * A129880 A137954 A028283
Adjacent sequences: A033294 A033295 A033296 * A033298 A033299 A033300
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu)
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EXTENSIONS
| Corrected Hankel transform Paul Barry (pbarry(AT)wit.ie), Nov 04 2009
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