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Search: id:A101409
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| A101409 |
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Triangle read by rows: T(n,k) is the number of noncrossing trees with n edges in which the leftmost leaf is at level k. |
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1
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| 1, 1, 2, 3, 5, 4, 12, 19, 16, 8, 55, 85, 73, 44, 16, 273, 416, 361, 234, 112, 32, 1428, 2156, 1883, 1269, 680, 272, 64, 7752, 11628, 10200, 7043, 4016, 1856, 640, 128, 43263, 64581, 56829, 39897, 23665, 11864, 4848, 1472, 256, 246675, 366850, 323587, 229936, 140161, 74050, 33360, 12256, 3328, 512
(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 1 and row sums yield the ternary numbers (A001764).
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FORMULA
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T(n, k)=sum([(k+i)/(2*n-k+i)]binomial(k-1, i)binomial(3n-2k+i-1, n-k), i=0..k-1). G.f.=tzg/(1-tzg-tzg^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(2,1)=1 and T(2,1)=2 because the noncrossing trees with 2 edges are /\, /_ and _\.
Triangle starts:
1;
1,2;
3,5,4;
12,19,16,8;
55,85,73,44,16;
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MAPLE
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T:=proc(n, k) if k=1 then binomial(3*n-3, n-1)/(2*n-1) elif k<=n then sum(((k+i)/(2*n-k+i))*binomial(k-1, i)*binomial(3*n-2*k+i-1, n-k), i=0..k-1) else 0 fi end: for n from 1 to 10 do seq(T(n, k), k=1..n) od; # yields sequence in triangular form
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CROSSREFS
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Cf. A001764.
Sequence in context: A138153 A023395 A101428 this_sequence A131401 A061446 A107476
Adjacent sequences: A101406 A101407 A101408 this_sequence A101410 A101411 A101412
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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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