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A124431
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a(n) = Sum_{k=0..n} 2^k*C([(n+k)/2],k)*C([(n+k+1)/2],k)) where [x]=floor(x).
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1
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1, 3, 9, 29, 97, 331, 1145, 4001, 14089, 49915, 177713, 635293, 2278841, 8198227, 29567729, 106872961, 387038993, 1404052659, 5101219929, 18559193245, 67605310097, 246541193883, 899999057385, 3288522934433, 12026324883865
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,2
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COMMENTS
| This is the inverse Motzkin transform of A026378 assuming offset 1 here. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 07 2009]
Hankel transform is Somos-4 variant A162547. [From Paul Barry (pbarry(AT)wit.ie], Jan 9 2011]
a(n) is the number of peakless Motzkin paths of length n in which the (1,0)-steps at level 0 come in 3 colors and those at a higher level come in 2 colors. Example: a(3)=29 because, denoting U=(1,1), H=(1,0), and D=(1,-1), we have 3^3 = 27 paths of shape HHH and 2 paths of shape UHD. [Emeric Deutsch, May 3 2011]
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FORMULA
| a(n) = Sum_{k=0..n} 2^k*A124428(n+k,k).
Conjecture: G.f.:-1/2*(1-4*x+x^2-((x^2+1)*(1-4*x+x^2))^(1/2))/x/(1-4*x+x^2) [From Maksym Voznyy (voznyy(AT)mail.ru), Aug 12 2009]
G.f.: (1/(1-4*x+x^2))*c(-x/(1-4*x+x^2)), c(x) the g.f. of A000108. [From Paul Barry (pbarry(AT}wit.ie), Jan 9 2011]
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PROG
| (PARI) a(n)=sum(k=0, n, 2^k*binomial((n+k)\2, k)*binomial((n+k+1)\2, k))
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CROSSREFS
| Cf. A124428.
Sequence in context: A071736 A148938 A082306 * A071740 A081696 A148939
Adjacent sequences: A124428 A124429 A124430 * A124432 A124433 A124434
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KEYWORD
| nonn
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AUTHOR
| Paul D. Hanna (pauldhanna(AT)juno.com), Oct 31 2006
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