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A094322
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n having k pyramids.
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0
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1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 4, 3, 3, 3, 1, 13, 11, 7, 6, 4, 1, 42, 37, 23, 14, 10, 5, 1, 139, 122, 78, 43, 25, 15, 6, 1, 470, 408, 262, 145, 75, 41, 21, 7, 1, 1616, 1390, 887, 494, 251, 124, 63, 28, 8, 1, 5632, 4810, 3048, 1694, 864, 414, 196, 92, 36, 9, 1, 19852, 16857, 10622
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OFFSET
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0,9
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COMMENTS
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A pyramid in a Dyck path is a factor of the form U^j D^j (j>0), starting at the x-axis. Here U=(1,1) and D(1,-1). This definition differs from the one in A091866. Column k=0 is A082989. Row sums are the Catalan numbers (A000108).
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LINKS
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Table of n, a(n) for n=0..68.
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FORMULA
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G.f.=G=G(t, z)=(1-z)/(1-zC+z^2*C -tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(3,2)=2 because there are two Dyck paths of semilength 3 having 2 pyramids: (UD)(UUDD) and (UUDD)(UD) (pyramids shown between parentheses).
Triangle begins:
[1];[0, 1];[0, 1, 1];[1, 1, 2, 1];[4, 3, 3, 3, 1];[13, 11, 7, 6, 4, 1];[42, 37, 23, 14, 10, 5, 1];
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=(1-z)/(1-z*C+z^2*C-t*z): Gserz:=simplify(series(G, z=0, 16)): P[0]:=1: for n from 1 to 14 do P[n]:=sort(coeff(Gserz, z^n)) od: seq([subs(t=0, P[n]), seq(coeff(P[n], t^k), k=1..n)], n=0..14);
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CROSSREFS
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Cf. A082989, A000108.
Sequence in context: A159931 A159755 A215534 * A136757 A134599 A117235
Adjacent sequences: A094319 A094320 A094321 * A094323 A094324 A094325
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KEYWORD
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nonn,tabl
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AUTHOR
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Emeric Deutsch, Jun 03 2004
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STATUS
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approved
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