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Search: id:A097862
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| A097862 |
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Triangle read by rows: T(n,k) is the number of Motzkin paths of length n and height k (n>=0, k>=0). |
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+0
1
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| 1, 1, 1, 1, 1, 3, 1, 7, 1, 1, 15, 5, 1, 31, 18, 1, 1, 63, 56, 7, 1, 127, 161, 33, 1, 1, 255, 441, 129, 9, 1, 511, 1170, 453, 52, 1, 1, 1023, 3036, 1485, 242, 11, 1, 2047, 7753, 4644, 990, 75, 1, 1, 4095, 19565, 14040, 3718, 403, 13, 1, 8191, 48930, 41392, 13145, 1872, 102
(list; graph; listen)
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OFFSET
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0,6
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COMMENT
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Row sums are the Motzkin numbers (A001006).
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FORMULA
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The g.f. for column k is z^(2k)/[P_k*P_{k+1}], where the polynomials P_k are defined by P_0=1, P_1=1-z, P_k=(1-z)P_{k-1}-z^2*P_{k-2}.
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EXAMPLE
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Triangle begins:
1;
1;
1,1;
1,3;
1,7,1;
1,15,5;
1,31,18,1;
1,63,56,7;
1,127,161,33,1;
Row n contains 1+floor(n/2) terms.
T(5,2)=5 counts HUUDD, UUDDH, UUDHD, UHUDD and UUHDD, where U=(1,1), H=(1,0) and D=(1,-1).
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MAPLE
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P[0]:=1: P[1]:=1-z: for n from 2 to 10 do P[n]:=sort(expand((1-z)*P[n-1]-z^2*P[n-2])) od: for k from 0 to 8 do h[k]:=series(z^(2*k)/P[k]/P[k+1], z=0, 20) od: a:=proc(n, k) if k=0 then 1 elif n=0 then 0 else coeff(h[k], z^n) fi end: seq(seq(a(n, k), k=0..floor(n/2)), n=0..15);
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CROSSREFS
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Cf. A001006.
Sequence in context: A143470 A114580 A097229 this_sequence A097612 A136011 A021991
Adjacent sequences: A097859 A097860 A097861 this_sequence A097863 A097864 A097865
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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), Sep 01 2004
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