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Search: id:A109077
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| A109077 |
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Triangle read by rows: T(n,k) is the number of symmetric Dyck paths of semilength n and having k hills (i.e. peaks at level 1). |
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+0
2
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| 1, 0, 1, 1, 0, 1, 2, 0, 0, 1, 4, 0, 1, 0, 1, 6, 1, 2, 0, 0, 1, 13, 0, 5, 0, 1, 0, 1, 22, 2, 6, 2, 2, 0, 0, 1, 46, 0, 16, 0, 6, 0, 1, 0, 1, 80, 6, 24, 4, 6, 3, 2, 0, 0, 1, 166, 0, 58, 0, 19, 0, 7, 0, 1, 0, 1, 296, 18, 90, 13, 26, 6, 6, 4, 2, 0, 0, 1, 610, 0, 211, 0, 71, 0, 22, 0, 8, 0, 1, 0, 1, 1106
(list; table; graph; listen)
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OFFSET
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0,7
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COMMENT
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Column 0 yields A109078. T(2n,1)=0, T(2n-1,1)=A000957(n) (the Fine numbers).
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FORMULA
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G.f.=2[1+(t-1)z(1-2z)+q(1-z+tz)]/[(1-2z+q)(1+2z^2-2t^2*z^2+q)], where q=sqrt(1-4z^2).
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EXAMPLE
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T(5,2)=2 because we have uduududdud and uduuudddud, where u=(1,1), d=(1,-1).
Triangle begins:
1;
0,1;
1,0,1;
2,0,0,1;
4,0,1,0,1;
6,1,2,0,0,1;
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MAPLE
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G:=-2*(z+z*sqrt(1-4*z^2)-2*z^2-z*t-1-sqrt(1-4*z^2)+2*z^2*t-z*t*sqrt(1-4*z^2))/(-1-sqrt(1-4*z^2)+2*z)/(-1-sqrt(1-4*z^2)-2*z^2+2*z^2*t^2): Gser:=simplify(series(G, z=0, 17)): P[0]:=1: for n from 1 to 13 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 13 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A109078, A000957.
Adjacent sequences: A109074 A109075 A109076 this_sequence A109078 A109079 A109080
Sequence in context: A123634 A091866 A111146 this_sequence A137585 A072458 A067310
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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), Jun 17 2005
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