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Search: id:A132888
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| A132888 |
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Triangle read by rows: T(n,k) is the number of paths in the first quadrant, from (0,0) to (n,0), consisting of steps U=[1,1], D[1,-1], h=(1,0), and H=(2,0), having height k (0<=k<=floor(n/2)). |
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+0
1
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| 1, 1, 2, 1, 3, 3, 5, 10, 1, 8, 27, 5, 13, 72, 23, 1, 21, 183, 86, 7, 34, 459, 302, 40, 1, 55, 1134, 994, 185, 9, 89, 2782, 3158, 778, 61, 1, 144, 6786, 9742, 3027, 332, 11, 233, 16498, 29450, 11196, 1607, 86, 1, 377, 40014, 87602, 39795, 7117, 535, 13
(list; graph; listen)
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OFFSET
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0,3
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COMMENT
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Row n has 1+floor(n/2) terms. T(n,0)=A000045(n+1) (the Fibonacci numbers). Row sums yield A128720.
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FORMULA
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G.f. of column k is g[k]=G[k]-G[k-1], where G[0]:=1/(1-z-z^2), G[k]=1/(1-z-z^2-z^2*G[k-1]).
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EXAMPLE
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T(3,1)=3 because we have hUD, UhD, and UDh.
Triangle starts:
1;
1;
2,1;
3,3;
5,10,1;
8,27,5;
13,72,23,1;
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MAPLE
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G[0]:=1/(1-z-z^2): for n to 13 do G[n]:=simplify(1/(1-z-z^2-z^2*G[n-1])) end do: g[0]:=G[0]: for n to 13 do g[n]:=simplify(G[n]-G[n-1]) end do: for n from 0 to 13 do gser[n]:=series(g[n], z=0, 40) end do: for n from 0 to 13 do seq(coeff(gser[k], z, n), k=0..floor((1/2)*n)) end do;
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CROSSREFS
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Cf. A000045, A128720.
Sequence in context: A059876 A095354 A132883 this_sequence A124774 A056610 A096373
Adjacent sequences: A132885 A132886 A132887 this_sequence A132889 A132890 A132891
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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 05 2007
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