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Search: id:A097611
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| A097611 |
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Triangle read by rows: T(n,k) is number of Motzkin paths of length n having k peaks at height 1. |
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+0
1
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| 1, 1, 1, 1, 2, 2, 5, 3, 1, 12, 6, 3, 29, 15, 6, 1, 72, 38, 13, 4, 183, 96, 33, 10, 1, 473, 246, 87, 24, 5, 1239, 641, 229, 63, 15, 1, 3282, 1692, 606, 172, 40, 6, 8777, 4512, 1620, 470, 110, 21, 1, 23665, 12136, 4370, 1284, 311, 62, 7, 64261, 32887, 11874, 3523, 880, 180
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OFFSET
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0,5
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COMMENT
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Row sums are the Motzkin numbers (A001006). Column 0 is A089372.
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FORMULA
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G.f.= 2/[1-z+2z^2-2tz^2+sqrt(1-2z-3z^2)].
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EXAMPLE
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Triangle begins:
1;
1;
1,1;
2,2;
5,3,1;
12,6,3;
Row n has 1+floor(n/2) terms.
T(5,2)=3 because H(UD)(UD), (UD)H(UD), (UD)(UD)H are the only Motzkin paths of length 5 with 2 peaks at height 1 (shown between parentheses); here U=(1,1),
H=(1,0) and D=(1,-1).
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MAPLE
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G := 2/(1-z+sqrt(1-2*z-3*z^2)+2*z^2-2*z^2*t): Gser:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 15 do P[n]:=sort(coeff(Gser, z^n)) od: seq(seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)), n=0..15);
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CROSSREFS
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Cf. A001006, A089372.
Sequence in context: A085483 A038041 A097891 this_sequence A135376 A132850 A076561
Adjacent sequences: A097608 A097609 A097610 this_sequence A097612 A097613 A097614
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 30 2004
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