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A143398
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Triangle T(n,k) = number of forests of labeled rooted trees of height at most 1, with n labels, where each root contains k labels, n>=0, 0<=k<=n.
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9
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1, 0, 1, 0, 3, 1, 0, 10, 3, 1, 0, 41, 9, 4, 1, 0, 196, 40, 10, 5, 1, 0, 1057, 210, 30, 15, 6, 1, 0, 6322, 1176, 175, 35, 21, 7, 1, 0, 41393, 7273, 1176, 105, 56, 28, 8, 1, 0, 293608, 49932, 7084, 756, 126, 84, 36, 9, 1, 0, 2237921, 372060, 42120, 6510, 378, 210, 120, 45, 10, 1
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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LINKS
| Index entries for sequences related to rooted trees
Alois P. Heinz, Table of n, a(n) for n = 0..10010
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FORMULA
| T(n,k) = n! * Sum_{i=0..u(n,k)} i^(n-k*i)/((n-k*i)!*i!*k!^i) with u(n,k) = 0 if k=0 and u(n,k) = floor(n/k) else.
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EXAMPLE
| T(4,2) = 9: 3->{1,2}<-4, 2->{1,3}<-4, 2->{1,4}<-3, 1->{2,3}<-4, 1->{2,4}<-3, 1->{3,4}<-2, {1,2}{3,4}, {1,3}{2,4}, {1,4}{2,3}.
Triangle begins:
1;
0, 1;
0, 3, 1;
0, 10, 3, 1;
0, 41, 9, 4, 1;
0, 196, 40, 10, 5, 1;
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MAPLE
| u:= (n, k)-> `if` (k=0, 0, floor(n/k)):
T:= (n, k)-> n! *add (i^(n-k*i)/ ((n-k*i)! *i! *k!^i), i=0..u(n, k)):
seq (seq (T(n, k), k=0..n), n=0..12);
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CROSSREFS
| Columns k=0-2: A000007, A000248, A133189. Diagonal: A000012. See also A000142, A143406 for row sums.
Sequence in context: A135871 A126178 A094753 * A202995 A191578 A067176
Adjacent sequences: A143395 A143396 A143397 * A143399 A143400 A143401
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KEYWORD
| nonn,tabl
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AUTHOR
| Alois P. Heinz (heinz(AT)hs-heilbronn.de), Aug 12 2008
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