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1, 4, 14, 44, 134, 400, 1184, 3488, 10253, 30108, 88386, 259492, 762085, 2239120, 6582280, 19360432, 56976859, 167774428, 494301778, 1457104948, 4297477252, 12680944960, 37436553544, 110569987344, 326713395019, 965775778420
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| a(n)=number of peaks at even height in all Motzkin paths of length n+3. Example: a(2)=4 because in the 21 Motzkin paths of length 5 we have alltogether 4 peaks at even height (shown between parentheses): HU(UD)D, U(UD)DH, U(UD)HD, UH(UD)D.
This is a kind of Motzkin transform of A121262 because the substitution x -> x*A001006(x) in the independent variable of the g.f. A121262(x) defines a sequence which is 1,0,0,0 followed by this sequence here. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Nov 08 2008]
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FORMULA
| G.f. = (1-2z-z^2)/[2z^3*(1-z)sqrt(1-2z-3z^2)]-1/(2z^3);
Conjecture: -(n-1)*(n+3)*a(n) +(n+2)*(3n-1)*a(n-1) +(n-1)*(n+1)*a(n-2) -3*n*(n+1)*a(n-3)=0. - R. J. Mathar, Nov 17 2011
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MAPLE
| ser:=series((1-2*z-z^2)/2/z^3/(1-z)/sqrt(1-2*z-3*z^2)-1/2/z^3, z=0, 32): seq(coeff(ser, z^n), n=1..28);
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CROSSREFS
| Cf. A014531.
Sequence in context: A000300 A005323 A027831 * A065835 A198643 A097137
Adjacent sequences: A097891 A097892 A097893 * A097895 A097896 A097897
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KEYWORD
| nonn
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2004
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