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A025266
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a(n) = a(1)*a(n-1) + a(2)*a(n-2) + ...+ a(n-1)*a(1) for n >= 4.
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1
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1, 1, 0, 1, 2, 6, 16, 45, 126, 358, 1024, 2954, 8580, 25084, 73760, 218045, 647670, 1932230, 5787520, 17398270, 52476700, 158765300, 481690080, 1465239250, 4467799212, 13653601116, 41812009216, 128290240180, 394338641416, 1214165174712
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| a(n+2)=number of Motzkin (2n)-paths whose longest plateau is of length n. A plateau is a sequence of contiguous flatsteps that is either the entire path or is of length >=1 and preceded by an up step and followed by a down step. Example: for n=3; a(5) counts UFFFDF and FUFFFD. - David Callan (callan(AT)stat.wisc.edu), Jul 15 2004
a(n) is the number of Motzkin paths of length n-2 having no (1,0)-steps at levels 0,2,4,... and having (1,0)-steps of two colors at levels 1,3,5,... . Example: a(7)=16 because, denoting U=(1,1), D=(1,-1), and H=(1,0), we have 2 paths of shape UDUHD, 2 paths of shape UHDUD, 2^3 = 8 paths of shape UHHHD, 2 paths of shape UHUDD, and 2 paths of shape UUDHD. [Emeric Deutsch, May 2 2011]
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FORMULA
| G.f.: (1-sqrt(1-4*x+8*x^3))/2 - Michael Somos, Jun 08, 2000.
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PROG
| (PARI) a(n)=polcoeff((1-sqrt(1-4*x+8*x^3+x*O(x^n)))/2, n)
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CROSSREFS
| Cf. A025264.
Sequence in context: A126285 A026163 A005717 * A074403 A151391 A166896
Adjacent sequences: A025263 A025264 A025265 * A025267 A025268 A025269
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KEYWORD
| nonn
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AUTHOR
| Clark Kimberling (ck6(AT)evansville.edu)
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