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A014300
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Number of nodes of odd outdegree in all ordered rooted (planar) trees with n edges.
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15
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1, 2, 7, 24, 86, 314, 1163, 4352, 16414, 62292, 237590, 909960, 3497248, 13480826, 52097267, 201780224, 783051638, 3044061116, 11851853042, 46208337584, 180383564228, 704961896036, 2757926215742, 10799653176704
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Also total number of blocks of odd size in all Catalan(n) possible noncrossing partitions of [n].
Convolution of the sequence of central binomial coefficients 1,2,6,20,70,... (A000984) and of the sequence of Fine numbers 1,0,1,2,6,18,... (A000957).
Row sums of A119307. - Paul Barry (pbarry(AT)wit.ie), May 13 2006
Hankel transform is A079935. [From Paul Barry (pbarry(AT)wit.ie), Jul 17 2009]
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REFERENCES
| Paul Barry, A Catalan Transform and Related Transformations on Integer Sequences, Journal of Integer Sequences, Vol. 8 (2005), Article 05.4.5.
N. Dershowitz and S. Zaks, Ordered trees and non-crossing partitions, Discrete Math., 62 (1986), 215-218.
E. Deutsch and L. Shapiro, A survey of the Fine numbers, Discrete Math., 241 (2001), 241-265.
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LINKS
| Index entries for sequences related to rooted trees
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FORMULA
| 2*binomial(2*n-1, n)/3 + A000957(n)/3;
Sum_{k=0..n} (-1)^(n-k)*binomial(n+k-1, k-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Aug 28 2002
G.f.: 2z/[1-4z+(1+2z)sqrt(1-4z)].
a(n)=sum(binomial(2n-2j-2, n-1), j=0..floor((n-1)/2)).
2*a(n) + a(n-1)=(3*n-1)*Catalan(n-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Dec 03 2004
a(n)=(-1)^n*sum(i=0, n, sum(j=n, 2*n, (-1)^(i+j)*binomial(j, i))) - Benoit Cloitre (benoit7848c(AT)orange.fr), Jun 18 2005
a(n)=sum{k=0..n, C(2k,n)} [offset 0]. - Paul Barry (pbarry(AT)wit.ie), May 13 2006
a(n)=sum{k=0..n, (-1)^(n-k)*C(n+k-1,k-1)}; - Paul Barry (pbarry(AT)wit.ie), Jul 18 2006
sum(igcd(binomial(2*j,n)),j=0..n). - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 25 2006
a(n)=sum{k=0..n, C(2n-k,n-k)*(1+(-1)^k)/2}=sum{k=0..n, C(n+k,k)(1+(-1)^(n-k))/2}. [From Paul Barry (pbarry(AT)wit.ie), Jul 17 2009]
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MAPLE
| a:=n->sum(igcd(binomial(2*j, n)), j=0..n): seq(a(n), n=0..23); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 25 2006
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MATHEMATICA
| Rest[CoefficientList[Series[2x/(1-4x+(1+2x)Sqrt[1-4x]), {x, 0, 40}], x]] (* From Harvey P. Dale, Apr 25 2011 *)
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CROSSREFS
| Cf. A059481.
Cf. A000957, A000984.
Sequence in context: A052986 A053368 A141753 * A128086 A131824 A150389
Adjacent sequences: A014297 A014298 A014299 * A014301 A014302 A014303
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KEYWORD
| nonn,nice,easy
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu)
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