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Search: id:A129165
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| A129165 |
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Triangle read by rows: T(n,k) is the number of skew Dyck paths of semilength n and having k base pyramids. A skew Dyck path is a path in the first quadrant which begins at the origin, ends on the x-axis, consists of steps U=(1,1)(up), D=(1,-1)(down) and L=(-1,-1)(left) so that up and left steps do not overlap. The length of the path is defined to be the number of its steps. A pyramid in a skew Dyck word (path) is a factor of the form u^h d^h, h being the height of the pyramid. A base pyramid is a pyramid starting on the x-axis. |
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+0
3
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| 1, 0, 1, 1, 1, 1, 5, 2, 2, 1, 19, 9, 4, 3, 1, 73, 37, 15, 7, 4, 1, 292, 147, 63, 24, 11, 5, 1, 1203, 598, 258, 100, 37, 16, 6, 1, 5065, 2497, 1067, 419, 152, 55, 22, 7, 1, 21697, 10633, 4507, 1762, 647, 224, 79, 29, 8, 1, 94274, 45980, 19379, 7528, 2765, 964, 322
(list; table; graph; listen)
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OFFSET
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0,7
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COMMENT
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Row sums yield A002212. T(n,0)=A129166(n). Sum(k*T(n,k),k=0..n)=A129167(n).
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REFERENCES
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E. Deutsch, E. Munarini and S. Rinaldi, Skew Dyck paths (in preparation).
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FORMULA
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G.f.=G(t,z)=(1-z)(1-z+zg)/[1-z(1-z)g-tz], where g=1+zg^2+z(g-1)=[1-z-sqrt(1-6z+5z^2)](2z).
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EXAMPLE
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T(3,1)=2 because we have (UD)UUDL and (UUUDDD) (the base pyramids are shown between parentheses).
Triangle starts:
1;
0,1;
1,1,1;
5,2,2,1;
19,9,4,3,1;
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MAPLE
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g:=(1-z-sqrt(1-6*z+5*z^2))/2/z: G:=(1-z)*(1-z+z*g)/(1-z*(1-z)*g-t*z): Gser:=simplify(series(G, z=0, 15)): for n from 0 to 11 do P[n]:=sort(expand(coeff(Gser, z, n))) od: for n from 0 to 11 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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Cf. A002212, A129166, A129167.
Sequence in context: A107719 A021661 A058841 this_sequence A081119 A119320 A070962
Adjacent sequences: A129162 A129163 A129164 this_sequence A129166 A129167 A129168
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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), Apr 04 2007
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