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Search: id:A132277
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| A132277 |
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Triangle read by rows: T(n,k) is number of paths in the first quadrant from (0,0) to (n,0) using steps U=(1,1), D=(1,-1), h=(1,0), and H=(2,0), having exactly k h-steps. |
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+0
3
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| 1, 0, 1, 2, 0, 1, 0, 5, 0, 1, 6, 0, 9, 0, 1, 0, 25, 0, 14, 0, 1, 22, 0, 66, 0, 20, 0, 1, 0, 129, 0, 140, 0, 27, 0, 1, 90, 0, 450, 0, 260, 0, 35, 0, 1, 0, 681, 0, 1210, 0, 441, 0, 44, 0, 1, 394, 0, 2955, 0, 2765, 0, 700, 0, 54, 0, 1, 0, 3653, 0, 9625, 0, 5642, 0, 1056, 0, 65, 0, 1
(list; table; graph; listen)
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OFFSET
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0,4
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COMMENT
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T(2n+1,0)=0; T(2n,0)=A006318(n) (the large Schroder numbers). Row sums yield A128720. Sum(k*T(n,k),k=0..n)=A106053(n+1).
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FORMULA
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G.f.=G=G(t,z) satisfies G = 1 + tzG + z^2*G + z^2*G^2 (see explicit expression at the Maple program).
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EXAMPLE
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T(4,2)=9 because we have hhH, hhUD, hHh, hUDh, Hhh, UDhh, hUhD, UhDh, and UhhD.
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MAPLE
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G:=((1-t*z-z^2-sqrt((1-2*z-t*z-z^2)*(1+2*z-t*z-z^2)))*1/2)/z^2: Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 11 do seq(coeff(P[n], t, j), j=0..n) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A006318, A128720, A106053.
Sequence in context: A067631 A123641 A134317 this_sequence A137286 A128890 A078924
Adjacent sequences: A132274 A132275 A132276 this_sequence A132278 A132279 A132280
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), Aug 26 2007
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