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Search: id:A106834
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| A106834 |
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Triangle read by rows: T(n, m) = number of painted forests on labeled vertex set [n] with m trees. Also number of painted forests with exactly n - m edges. |
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| 1, 1, 2, 3, 6, 3, 16, 30, 18, 4, 125, 220, 135, 40, 5, 1296, 2160, 1305, 420, 75, 6, 16807, 26754, 15750, 5180, 1050, 126, 7, 262144, 401408, 229824, 75460, 16100, 2268, 196, 8, 4782969, 7085880, 3949722, 1282176, 278775, 42336, 4410, 288, 9
(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 sums equal A101313 (Number of painted forests - exactly one of its trees is painted - on labeled vertex set [n].).
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LINKS
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Washington Bomfim, Illustration Of This Sequence.
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FORMULA
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T(n, m)= m * f(n, m), where f(n, m) = number of forests with n nodes and m labeled trees, A105599.
E.g.f.: y*B(x)*exp(y*B(x)), where B(x) is e.g.f. for A000272. - Vladeta Jovovic (vladeta(AT)Eunet.yu), May 24 2005
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EXAMPLE
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T(4, 3) = 18 because there are 18 such forests with 4 nodes and 3 trees. (See the illustration of this sequence).
Triangle begins
1,
1, 2,
3, 6, 3,
16, 30, 18, 4,
125, 220, 135, 40, 5,
1296, 2160, 1305, 420, 75, 6,
16807, 26754, 15750, 5180, 1050, 126, 7,
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MAPLE
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f:= proc(n, m) option remember; if n<0 then 0 elif n=m then 1 elif m<1 or m>n then 0 else add (binomial (n-1, j-1) *j^(j-2) *f(n-j, m-1), j=1..n-m+1) fi end: T:= (n, m)-> m*f(n, m): seq (seq (T(n, m), m=1..n), n=0..12); [From Alois P. Heinz (heinz(AT)hs-heilbronn.de), Sep 10 2008]
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CROSSREFS
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Cf. A101313, A105599, A106240.
Adjacent sequences: A106831 A106832 A106833 this_sequence A106835 A106836 A106837
Sequence in context: A019773 A109536 A101401 this_sequence A021427 A091834 A050210
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KEYWORD
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easy,nonn,tabl
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AUTHOR
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Washington Bomfim (webonfim(AT)bol.com.br), May 19 2005
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