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A120906
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Triangle read by rows: T(n,k) is the number of ternary words of length n on {0,1,2} having k drops (n>=0, k>=0). The drops of a ternary word on {0,1,2} are the subwords 10,20 and 21.
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2
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1, 3, 6, 3, 10, 16, 1, 15, 51, 15, 21, 126, 90, 6, 28, 266, 357, 77, 1, 36, 504, 1107, 504, 36, 45, 882, 2907, 2304, 414, 9, 55, 1452, 6765, 8350, 2850, 210, 1, 66, 2277, 14355, 25653, 14355, 2277, 66, 78, 3432, 28314, 69576, 58278, 16236, 1221, 12, 91, 5005
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OFFSET
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0,2
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COMMENTS
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Row n has 1+floor(2n/3) terms. Row sums are the powers of 3 (A000244). T(n,0)=A000217(n+1) (the triangular numbers). Sum(k*T(n,k),k>=0)=(n-1)*3^(n-1)=A036290(n-1).
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LINKS
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Table of n, a(n) for n=0..53.
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FORMULA
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G.f.=G(t,z)=1/[(1-z)^3-3tz^2+2tz^3-t^2*z^3].
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EXAMPLE
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T(5,3)=6 because we have 1/02/1/0, 2/02/1/0, 2/1/01/0, 2/1/02/0, 2/12/1/0 and 2/1/02/1, the middle points of the drops being indicated by /.
Triangle starts:
1;
3;
6,3;
10,16,1;
15,51,15;
21,126,90,6
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MAPLE
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G:=1/((1-z)^3-3*t*z^2+2*t*z^3-t^2*z^3): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(coeff(Gser, z^n)) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..floor(2*n/3)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000244, A000217, A036290.
Sequence in context: A129529 A128503 A210193 * A210201 A160899 A203491
Adjacent sequences: A120903 A120904 A120905 * A120907 A120908 A120909
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, Jul 15 2006
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STATUS
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approved
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