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Search: id:A007854
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| A007854 |
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G.f.: 1/(1-3*x*C) where C = (1/2-1/2*(1-4*x)^(1/2))/x = g.f. for Catalan numbers A000108. |
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14
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| 1, 3, 12, 51, 222, 978, 4338, 19323, 86310, 386250, 1730832, 7763550, 34847796, 156503064, 703149438, 3160160811, 14206181382, 63874779714, 287242041528, 1291872728826, 5810776384932, 26138647551564, 117587214581508
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Chains in rooted plane trees on n nodes.
The Hankel transform of the aerated sequence with g.f. 1/(1-3x^2c(x^2)) is also 3^n. In general, the expansions of 1/(1-k*x*c(x)) and 1/(1-k*x^2*c(x^2)) have Hankel transform k^n. - Paul Barry (pbarry(AT)wit.ie), Jan 20 2007
Binomial transform of A112657 . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 25 2007
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REFERENCES
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M. Klazar, Twelve countings with rooted plane trees, European Journal of Combinatorics 18 (1997), 195-210; Addendum, 18 (1997), 739-740.
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LINKS
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Index entries for sequences related to rooted trees
J.-C. Novelli and J.-Y. Thibon, Free quasi-symmetric functions of arbitrary level
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FORMULA
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a(n) = (9*a(n-1)-3*A000108(n-2))/2 = 3*A049027(n-1) = A067336(n-1)*3/2 = A049027(n-1)+A067336(n-1) = A067347(3, n-1). - Henry Bottomley (se16(AT)btinternet.com), Jan 16 2002
a(n) = Sum_{k>=0} A106566(n, k)*3^k . - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Aug 11 2005
The Hankel transform of this sequence is A000244 = [1, 3, 9, 27, 81, 243, 729, ...](powers of 3). - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Nov 26 2006
a(n)=sum{k=0..n, C(2n,n-k)(2k+1)2^k/(n+k+1)}; - Paul Barry (pbarry(AT)wit.ie), Jan 20 2007
a(n)= Sum{k, 0<=k<=n}A039599(n,k)*2^k. - Philippe DELEHAM (kolotoko(AT)wanadoo.fr), Sep 08 2007
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CROSSREFS
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Cf. A000108, A000984, A076035, A076036, A067347.
Sequence in context: A083314 A104268 A081704 this_sequence A110167 A064036 A125187
Adjacent sequences: A007851 A007852 A007853 this_sequence A007855 A007856 A007857
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KEYWORD
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nonn
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AUTHOR
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Martin Klazar (klazar(AT)kam.mff.cuni.cz)
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EXTENSIONS
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More terms from Henry Bottomley (se16(AT)btinternet.com), Jan 16 2002
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