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Search: id:A108263
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| A108263 |
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Triangle read by rows: T(n,k) is the number of short bushes with n edges and k branchnodes (i.e. nodes of outdegree at least two). A short bush is an ordered tree with no nodes of outdegree 1. |
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+0
7
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| 1, 0, 0, 1, 0, 1, 0, 1, 2, 0, 1, 5, 0, 1, 9, 5, 0, 1, 14, 21, 0, 1, 20, 56, 14, 0, 1, 27, 120, 84, 0, 1, 35, 225, 300, 42, 0, 1, 44, 385, 825, 330, 0, 1, 54, 616, 1925, 1485, 132, 0, 1, 65, 936, 4004, 5005, 1287, 0, 1, 77, 1365, 7644, 14014, 7007, 429, 0, 1, 90, 1925, 13650, 34398
(list; graph; listen)
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OFFSET
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0,9
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Riordan numbers (A005043). Column 3 yields A033275; column 4 yields A033276.
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REFERENCES
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F. R. Bernhart, Catalan, Motzkin, and Riordan numbers, Discr. Math., 204 (1999) 73-112.
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FORMULA
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G.f.=G=G(t, z) satisfies z(1+tz)G^2-(1+z)G+1=0.
T(n, k) = A086810(n-k, k) . - Philippe DELEHAM, May 30 2005
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EXAMPLE
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T(6,3)=5 because the only short bushes with 6 edges and 3 branchnodes are the five full binary trees with 6 edges.
Triangle begins:
1;
0;
0,1;
0,1;
0,1,2;
0,1,5;
0,1,9,5
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MAPLE
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G:=(1+z-sqrt((1-z)^2-4*t*z^2))/2/z/(1+t*z): Gser:=simplify(series(G, z=0, 18)): P[0]:=1: for n from 1 to 16 do P[n]:=coeff(Gser, z^n) od: for n from 0 to 16 do seq(coeff(t*P[n], t^k), k=1..1+floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A005043, A033275, A033276.
Columns : A000007, A000012, A000096, A033275, A033276, A033277, A033278, A033279; A05043 (row sums), A033282, A086810
Sequence in context: A106579 A016584 A112899 this_sequence A134433 A125183 A092583
Adjacent sequences: A108260 A108261 A108262 this_sequence A108264 A108265 A108266
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KEYWORD
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nonn,tabf
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AUTHOR
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Emeric Deutsch (deutsch(AT)duke.poly.edu), May 29 2005
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