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Search: id:A127541
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| A127541 |
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Triangle read by rows: T(n,k) is the number of ordered trees with n edges having k even-length branches starting at the root (0<=k<=n). |
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+0
3
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| 1, 1, 1, 1, 3, 2, 8, 5, 1, 24, 15, 3, 75, 46, 10, 1, 243, 148, 34, 4, 808, 489, 116, 16, 1, 2742, 1652, 402, 61, 5, 9458, 5678, 1408, 228, 23, 1, 33062, 19792, 4982, 847, 97, 6, 116868, 69798, 17783, 3138, 393, 31, 1, 417022, 248577, 63967, 11627, 1557, 143, 7
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OFFSET
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0,5
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COMMENT
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Row n has 1+floor(n/2) terms. Row sums are the Catalan numbers (A000108). T(n,0)=A000958(n-1). Sum(k*T(n,k),k=0..floor(n/2))=A127540(n-1).
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FORMULA
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G.f.=(1+z)/(1+z-z*C-tz^2*C), where C =[1-sqrt(1-4z)]/(2z) is the Catalan function.
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EXAMPLE
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T(2,0)=1 because we have the tree /\.
Triangle starts:
1;
1;
1,1;
3,2;
8,5,1;
24,15,3;
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MAPLE
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C:=(1-sqrt(1-4*z))/2/z: G:=(1+z)/(1+z-z*C-t*z^2*C): Gser:=simplify(series(G, z=0, 17)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: for n from 0 to 14 do seq(coeff(P[n], t, j), j=0..floor(n/2)) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A000958, A127538, A127540.
Sequence in context: A127300 A129199 A097018 this_sequence A053219 A083087 A072788
Adjacent sequences: A127538 A127539 A127540 this_sequence A127542 A127543 A127544
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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), Mar 01 2007
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