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A090345
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Number of Motzkin paths of length n with no level steps at even level.
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2
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1, 0, 1, 1, 3, 5, 12, 24, 55, 119, 272, 612, 1411, 3247, 7565, 17667, 41561, 98099, 232696, 553784, 1322813, 3169065, 7614583, 18342921, 44294991, 107200829, 259983346, 631718606, 1537737567, 3749440151, 9156561590, 22394270034
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| Hankel transform of a(n) is A000012. Hankel transform of a(n+1) is 0,-1,0,1,0,-1,0,... or -[x^n](x/(1+x^2)). Hankel transform of a(n+2) is A008619(n+1). [Paul Barry, Mar 23 2011]
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FORMULA
| G.f.: (1-z-sqrt(1-2*z-3*z^2+4*z^3))/(2*z^2).
G.f. A(x) satisfies A(x)=A(x/(x-1)). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 07 2004
Also (x*A)^2=(1-x)*(A-1). - Vladeta Jovovic (vladeta(AT)eunet.rs), Jul 07 2004
G.f.: 1/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-x^2/(1-x-x^2/(1-... (continued fraction). [From Paul Barry (pbarry(AT)wit.ie), Apr 08 2009]
G.f.: 1/(1-z/(1-z/(1-z/(...)))) where z=x^2/(1-x) (continued fraction); more generally g.f. C(x^2/(1-x)) where C(x) is the g.f. for the Catalan numbers (A000108). [Joerg Arndt, Mar 18 2011].
a(0)=1, a(n)=sum{k=0..floor(n/2), (k/(n-k))C(n-k,k)*A000108(k)}. [From Paul Barry (pbarry(AT)wit.ie), Jul 01 2009]
a(n)=sum{k=0..floor(n/2), C(n-k-1,n-2k)*A000108(k)}. [Paul Barry, Mar 23 2011]
The sequence starting with offset 1 = iterates of M*V, leftmost column. M = an infinite tridiagonal matrix with all 1's in the sub and superdiagonals and [0,1,0,1,0,1,0,1,...] as the main diagonal; and the rest zeros. V = vector [1,0,0,0,...]. [From Gary W. Adamson (qntmpkt(AT)yahoo.com), Jun 08 2011]
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EXAMPLE
| a(5)=5 because we have UHDUD, UDUHD, UHUDD, UUDHD and UHHHD, where U=(1,1),
D=(1,-1) and H=(1,0).
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CROSSREFS
| Cf. A001006.
First differences of A090344.
Sequence in context: A047761 A026786 A027246 * A185087 A186334 A151524
Adjacent sequences: A090342 A090343 A090344 * A090346 A090347 A090348
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Jan 28 2004
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