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Search: id:A102003
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| A102003 |
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Triangle read by rows: T(n,k) is the number of ordered trees with n edges and having k branches of odd length (n>=0, 0<=k<=n). |
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+0
2
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| 1, 0, 1, 1, 0, 1, 0, 3, 0, 2, 2, 0, 8, 0, 4, 0, 11, 0, 22, 0, 9, 5, 0, 45, 0, 61, 0, 21, 0, 41, 0, 166, 0, 171, 0, 51, 14, 0, 226, 0, 580, 0, 483, 0, 127, 0, 154, 0, 1050, 0, 1962, 0, 1373, 0, 323, 42, 0, 1070, 0, 4430, 0, 6496, 0, 3923, 0, 835, 0, 582, 0, 6005, 0, 17570, 0, 21184
(list; table; graph; listen)
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OFFSET
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0,8
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COMMENT
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Row n has n+1 terms. Column 0 yields the Catalan numbers (A000108) alternating with 0's. The diagonal entries are the Motzkin numbers (A001006). T(n,n-2)=A025566(n) for n>=2.
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REFERENCES
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E. Deutsch, Ordered trees with prescribed root degrees, node degrees, and branch lengths, Discrete Math., 282, 2004, 89-94.
J. Riordan, Enumeration of plane trees by branches and endpoints, J. Combin. Theory Ser. A 19, 1975, 214-222.
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FORMULA
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G.f.=G=G(t, z) satisfies z(t+z)G^2-(1+tz)G+1+tz=0.
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EXAMPLE
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T(3,3)=2 because we have (i) a tree with 3 edges hanging from the root and (ii) a tree with one edge hanging from the root, at the end of which 2 edges are hanging.
Triangle starts:
1;
0,1;
1,0,1;
0,3,0,2;
2,0,8,0,4;
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MAPLE
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G:=1/2/(z^2+t*z)*(t*z+1-sqrt(1-2*t*z-3*t^2*z^2-4*z^2-4*t*z^3)): Gserz:=simplify(series(G, z=0, 14)):P[0]:=1: for n from 1 to 12 do P[n]:=sort(expand(coeff(Gserz, z^n))) od: for n from 0 to 12 do seq(coeff(t*P[n], t^k), k=1..n+1) od; # yields the sequence in triangular form
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CROSSREFS
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Cf. A000108, A001006, A025566, A102004.
Sequence in context: A084196 A127913 A135991 this_sequence A004587 A104609 A087625
Adjacent sequences: A102000 A102001 A102002 this_sequence A102004 A102005 A102006
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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.eduandgessel(AT)brandeis.edu), Dec 23 2004
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