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A127156
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Triangle read by rows: T(n,k) is the number of Dyck paths of semilength n starting with exactly k consecutive pyramids. 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.
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1
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1, 0, 1, 0, 1, 1, 1, 1, 2, 1, 5, 2, 3, 3, 1, 19, 7, 5, 6, 4, 1, 67, 26, 12, 11, 10, 5, 1, 232, 93, 38, 23, 21, 15, 6, 1, 804, 325, 131, 61, 44, 36, 21, 7, 1, 2806, 1129, 456, 192, 105, 80, 57, 28, 8, 1, 9878, 3935, 1585, 648, 297, 185, 137, 85, 36, 9, 1, 35072, 13813, 5520
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OFFSET
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0,9
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COMMENTS
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Row sums yield the Catalan numbers (A000108). T(n,0)=A114277(n-3) for n>=3. Sum(k*T(n,k), k=0..n)=A014318(n-1) for n>=1.
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LINKS
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FORMULA
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G.f.=G(t,z)=(1-2z)C/(1-z-tz), where C=[1-sqrt(1-4z)]/(2z) is the Catalan function. T(n,k)=T(n-1,k)+T(n-1,k-1) for n,k>=1.
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EXAMPLE
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T(5,2)=5 because we have (UDUD)UUDUDD, (UUDDUUUDDD), (UUUDDDUUDD), (UDUUUUDDDD) and (UUUUDDDDUD) (the initial 2 pyramids are shown between parentheses).
Triangle starts:
1;
0,1;
0,1,1;
1,1,2,1;
5,2,3,3,1;
19,7,5,6,4,1;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=(1-2*z)*C/(1-z-t*z): Gser:=simplify(series(G, z=0, 15)): P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gser, z^n))) od: for n from 0 to 12 do seq(coeff(P[n], t, j), j=0..n) od; # yields sequence in triangular form
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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