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A191310
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Triangle read by rows: T(n,k) is the number of dispersed Dyck paths (i.e., Motzkin paths with no (1,0) steps at positive heights) of length n having k up-steps starting at level 0.
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1
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1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 6, 3, 1, 10, 8, 1, 1, 14, 16, 4, 1, 23, 32, 13, 1, 1, 32, 56, 32, 5, 1, 55, 102, 74, 19, 1, 1, 78, 170, 152, 55, 6, 1, 143, 302, 307, 144, 26, 1, 1, 208, 498, 580, 336, 86, 7, 1, 405, 890, 1102, 748, 251, 34, 1, 1, 602, 1478, 2004, 1564, 652, 126, 8, 1, 1228, 2691, 3714, 3200, 1587, 405, 43, 1
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OFFSET
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0,6
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COMMENTS
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Row n has 1 + floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
Sum_{k>=0} k*T(n,k) = A093387(n+1).
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LINKS
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Table of n, a(n) for n=0..80.
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FORMULA
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G.f.: G(t,z) = 2/(2-2*z-t*(1-sqrt(1-4*z^2))).
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EXAMPLE
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T(5,2)=3 because we have HUDUD, UDHUD, and UDUDH, where U=(1,1), D=(1,-1), H=(1,0).
Triangle starts:
1;
1;
1, 1;
1, 2;
1, 4, 1;
1, 6, 3;
1, 10, 8, 1;
1, 14, 16, 4;
1, 23, 32, 13, 1;
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MAPLE
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G := 2/(2-2*z-t*(1-sqrt(1-4*z^2))): Gser := simplify(series(G, z = 0, 20)): for n from 0 to 16 do P[n] := sort(coeff(Gser, z, n)) end do: for n from 0 to 16 do seq(coeff(P[n], t, k), k = 0 .. floor((1/2)*n)) end do; # yields sequence in triangular form
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CROSSREFS
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Cf. A001405, A093387.
Sequence in context: A130313 A247073 A124428 * A124845 A191392 A127625
Adjacent sequences: A191307 A191308 A191309 * A191311 A191312 A191313
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch, May 30 2011
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STATUS
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approved
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