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Search: id:A126219
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| A126219 |
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Triangle read by rows: T(n,k) is the number of binary trees (i.e. a rooted tree where each vertex has either 0,1, or 2 children; and, when only one child is present, it is either a right child or a left child ) with n edges and k pairs of adjacent vertices of outdegree 2. |
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+0
2
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| 1, 2, 5, 14, 40, 2, 116, 16, 344, 80, 5, 1040, 340, 50, 3188, 1360, 300, 14, 9880, 5264, 1484, 168, 30912, 19880, 6776, 1176, 42, 97520, 73728, 29568, 6608, 588, 309856, 269952, 124656, 33600, 4704, 132, 990656, 979264, 511584, 161280, 29544, 2112
(list; graph; listen)
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OFFSET
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0,2
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COMMENT
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Row n has floor(n/2) terms (n>=2). Row sums are the Catalan numbers (A000108). T(n,0)=A126220(n). Sum(k*T(n,k),k=0..floor(n/2)-1)=2*binom(2n-2,n-4)=2*A002696(n-1) (n>=4).
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FORMULA
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G.f.=G=G(t,z) satisfies G=1+2zG+z^2*[1+2zG+t(G-2zG-1)]^2 (see the Maple program for the explicit expression).
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EXAMPLE
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Triangle starts:
1;
2;
5;
14;
40,2;
116,16;
344,80,5;
1040,340,50;
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MAPLE
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G:=1/2*(1-4*z^3*t^2-4*z^3-2*z^2*t+8*z^3*t-2*z+2*z^2*t^2-sqrt(1-8*z^3+4*z^2-4*z^2\ *t-4*z+8*z^3*t))/z^2/(2*z*t-t-2*z)^2: Gser:=simplify(series(G, z=0, 18)): for n from 0 to 14 do P[n]:=sort(coeff(Gser, z, n)) od: 1; 2; for n from 2 to 14 do seq(coeff(P[n], t, j), j=0..floor(n/2)-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A000108, A126220, A002696.
Sequence in context: A148317 A148318 A148319 this_sequence A111110 A111109 A081908
Adjacent sequences: A126216 A126217 A126218 this_sequence A126220 A126221 A126222
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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), Dec 25 2006, Aug 17 2008
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