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A132885
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Triangle read by rows: T(n,k) is the number of paths in the right-half plane from (0,0) to (n,0), consisting of steps U=(1,1), D=(1,-1), h=(1,0) and H=(2,0), having k H=(2,0) steps (0<=k<=floor(n/2)).
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0
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1, 1, 3, 1, 7, 2, 19, 9, 1, 51, 28, 3, 141, 95, 18, 1, 393, 306, 70, 4, 1107, 987, 285, 30, 1, 3139, 3144, 1071, 140, 5, 8953, 9963, 3948, 665, 45, 1, 25653, 31390, 14148, 2856, 245, 6, 73789, 98483, 49815, 11844, 1330, 63, 1, 212941, 307836, 172645, 47160
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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COMMENTS
| Row n has 1+floor(n/2) terms. T(n,0)=A002426(n) (the central trinomial coefficients). T(n,1)=A109188(n-1). Row sums yield A059345. See A132280 for the same statistic on paths restricted to the first quadrant.
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FORMULA
| G.f.=1/sqrt((1+z-tz^2)((1-3z-tz^2)).
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EXAMPLE
| Triangle starts:
1;
1;
3,1;
7,2;
19,9,1;
51,28,3;
141,95,18,1;
T(4,1)=9 because we have hhH, hHh, Hhh, HUD, UDH, UHD, HDU, DUH and DHU.
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MAPLE
| G:=1/sqrt((1+z-t*z^2)*(1-3*z-t*z^2)): Gser:=simplify(series(G, z=0, 18)): for n from 0 to 13 do P[n]:=sort(coeff(Gser, z, n)) end do: for n from 0 to 13 do seq(coeff(P[n], t, j), j=0..floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
| Cf. A002426, A109188, A059345, A132280.
Sequence in context: A065259 A065289 A065265 * A059090 A133115 A104797
Adjacent sequences: A132882 A132883 A132884 * A132886 A132887 A132888
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KEYWORD
| nonn,tabf
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AUTHOR
| Emeric Deutsch (deutsch(AT)duke.poly.edu), Sep 03 2007
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