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A191318
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Triangle read by rows: T(n,k) is the number of dispersed Dyck paths of length n (i.e., Motzkin paths of length n with no (1,0) steps at positive heights) having pyramid weight equal to k.
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1
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1, 1, 1, 1, 1, 2, 1, 3, 2, 1, 4, 5, 1, 5, 10, 4, 1, 6, 16, 12, 1, 7, 24, 30, 8, 1, 8, 33, 56, 28, 1, 9, 44, 98, 84, 16, 1, 10, 56, 152, 179, 64, 1, 11, 70, 228, 358, 224, 32, 1, 12, 85, 320, 618, 536, 144, 1, 13, 102, 440, 1030, 1206, 576, 64, 1, 14, 120, 580, 1580, 2292, 1528, 320, 1, 15, 140, 754, 2370, 4202, 3820, 1440, 128
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OFFSET
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0,6
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COMMENTS
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A pyramid in a dispersed Dyck path is a factor of the form U^h D^h, h being the height of the pyramid and U=(1,1), D=(1,-1). A pyramid in a dispersed Dyck path w is maximal if, as a factor in w, it is not immediately preceded by a U and immediately followed by a D. The pyramid weight of a dispersed Dyck path is the sum of the heights of its maximal pyramids.
Row n has 1 + floor(n/2) entries.
Sum of entries in row n is binomial(n, floor(n/2)) = A001405(n).
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LINKS
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FORMULA
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T(n,0) = 1;
T(n,1) = n-1 (n>=1).
G.f.: G=G(t,z) satisfies z*(1-z)*(z-1+2*t*z^2)*G^2 + (1-z)*(z-1+2*t*z^2)*G+1-t*z^2=0.
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EXAMPLE
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T(6,2)=10 because we have HH(UD)(UD), HH(UUDD), H(UD)H(UD), H(UD)(UD)H, H(UUDD)H, (UD)HH(UD), (UD)H(UD)H, (UD)(UD)HH, (UUDD)HH, and U(UD)(UD)D, where U=(1,1), D=(1,-1), H=(1,0); the maximal pyramids are shown between parentheses.
Triangle starts:
1;
1;
1, 1;
1, 2;
1, 3, 2;
1, 4, 5;
1, 5, 10, 4;
1, 6, 16, 12;
1, 7, 24, 30, 8;
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MAPLE
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a := (z-1)*(2*t*z^2+z-1): c := -1+t*z^2: eq := a*z*G^2+a*G+c: f := RootOf(eq, G): fser := simplify(series(f, z = 0, 20)): for n from 0 to 16 do P[n] := sort(expand(coeff(fser, 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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KEYWORD
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nonn,tabf
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AUTHOR
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STATUS
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approved
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