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A005554
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Sum of successive Motzkin numbers.
(Formerly M0801)
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7
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1, 2, 3, 6, 13, 30, 72, 178, 450, 1158, 3023, 7986, 21309, 57346, 155469, 424206, 1164039, 3210246, 8893161, 24735666, 69051303, 193399578, 543310782, 1530523638, 4322488212, 12236130298, 34713220977, 98677591278
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| The Donaghey reference shows that a(n) is the number of n-vertex binary trees such that for each non-root vertex that is incident to exactly two edges, these two edges have opposite slope. It also notes that these trees correspond to Dyck n-paths (A000108) containing no DUDUs and no subpaths of the form UUPDD with P a nonempty Dyck path. For example, a(3)=3 counts UUDUDD, UDUUDD, UUDDUD. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
Hankel transform of the sequence starting with 2 appears to be 3, 4, 5, 6, 7,... Gary W. Adamson (qntmpkt(AT)yahoo.com), May 27 2011.
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REFERENCES
| R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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FORMULA
| Inverse binomial transform of A014138: (1, 3, 8, 22, 64, 196,...). - Gary W. Adamson (qntmpkt(AT)yahoo.com), Nov 23 2007
(n + 1)*a(n) = 2*n*a(n - 1) + (3*n - 9)*a(n - 2).
G.f.: (x+x^2)*M(x) where M(x)=(1 - x - (1 - 2*x - 3*x^2)^(1/2))/(2*x^2) is the g.f. for the Motzkin numbers A001006. - David Callan (callan(AT)stat.wisc.edu), Sep 25 2006
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CROSSREFS
| Enumerates the branch-reduced trees encoded by A080981. Cf. A001006.
First differences are in A102071.
Cf. A014138.
Sequence in context: A079512 A052937 A174191 * A077212 A076836 A202086
Adjacent sequences: A005551 A005552 A005553 * A005555 A005556 A005557
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KEYWORD
| nonn
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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EXTENSIONS
| More terms from James A. Sellers (sellersj(AT)math.psu.edu), Jul 10 2000
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