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Search: id:A101401
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| A101401 |
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost child of the root has degree k. |
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+0
1
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| 1, 1, 2, 3, 6, 3, 12, 24, 15, 4, 55, 110, 75, 28, 5, 273, 546, 390, 168, 45, 6, 1428, 2856, 2100, 980, 315, 66, 7, 7752, 15504, 11628, 5712, 2040, 528, 91, 8, 43263, 86526, 65835, 33516, 12825, 3762, 819, 120, 9, 246675, 493350, 379500, 198352, 79695, 25410, 6370, 1200, 153, 10
(list; table; graph; listen)
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OFFSET
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1,3
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COMMENT
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Row n contains n terms. Column 0 and row sums yield the ternary numbers (A001764).
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FORMULA
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T(n, k)=[(k+1)(2k+1)/(3n-k-2)]binomial(3n-k-2, 2n-1). G.f.=zg/(1-tzg^2)^2, where g=1+zg^3 is the g.f. of the ternary numbers (A001764).
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EXAMPLE
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T(2,0)=1 and T(2,1)=2 because the noncrossing trees with 2 edges are /\, |_ and _|.
Triangle starts:
1;
1,2;
3,6,3;
12,24,15,4;
55,110,75,28,5;
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MAPLE
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T:=proc(n, k) if n=1 and k=1 then 0 elif k<=n then (k+1)*(2*k+1)*binomial(3*n-k-2, 2*n-1)/(3*n-k-2) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=0..n-1) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001764.
Sequence in context: A129915 A019773 A109536 this_sequence A106834 A021427 A091834
Adjacent sequences: A101398 A101399 A101400 this_sequence A101402 A101403 A101404
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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), Jan 15 2005
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