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A215771
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Number T(n,k) of undirected labeled graphs on n nodes with exactly k cycle graphs as connected components; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
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15
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1, 0, 1, 0, 1, 1, 0, 1, 3, 1, 0, 3, 7, 6, 1, 0, 12, 25, 25, 10, 1, 0, 60, 127, 120, 65, 15, 1, 0, 360, 777, 742, 420, 140, 21, 1, 0, 2520, 5547, 5446, 3157, 1190, 266, 28, 1, 0, 20160, 45216, 45559, 27342, 10857, 2898, 462, 36, 1, 0, 181440, 414144, 427275, 264925, 109935, 31899, 6300, 750, 45, 1
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OFFSET
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0,9
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COMMENTS
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Also the Bell transform of A001710. For the definition of the Bell transform see A264428 and the links given there. - Peter Luschny, Jan 21 2016
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LINKS
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EXAMPLE
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T(4,1) = 3: .1-2. .1 2. .1-2.
. .| |. .|X|. . X .
. .3-4. .3 4. .3-4.
.
T(4,2) = 7: .1 2. .1-2. .1 2. o1 2. .1 2o .1-2. .1-2.
. .| |. . . . X . . /|. .|\ . . \|. .|/ .
. .3 4. .3-4. .3 4. .3-4. .3-4. o3 4. .3 4o
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T(4,3) = 6: .1 2o .1-2. o1 2. o1 2o o1 2. .1 2o
. .| . . . . |. . . . / . . \ .
. .3 4o o3 4o o3 4. .3-4. .3 4o o3 4.
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T(4,4) = 1: o1 2o
. . .
. o3 4o
Triangle T(n,k) begins:
1;
0, 1;
0, 1, 1;
0, 1, 3, 1;
0, 3, 7, 6, 1;
0, 12, 25, 25, 10, 1;
0, 60, 127, 120, 65, 15, 1;
0, 360, 777, 742, 420, 140, 21, 1;
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MAPLE
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T:= proc(n, k) option remember; `if`(k<0 or k>n, 0, `if`(n=0, 1,
add(binomial(n-1, i)*T(n-1-i, k-1)*ceil(i!/2), i=0..n-k)))
end:
seq(seq(T(n, k), k=0..n), n=0..12);
# Alternatively, with the function BellMatrix defined in A264428:
BellMatrix(n -> `if`(n<2, 1, n!/2), 8); # Peter Luschny, Jan 21 2016
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MATHEMATICA
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t[n_, k_] := t[n, k] = If[k < 0 || k > n, 0, If[n == 0, 1, Sum[Binomial[n-1, i]*t[n-1-i, k-1]*Ceiling[i!/2], {i, 0, n-k}]]]; Table[Table[t[n, k], {k, 0, n}], {n, 0, 12}] // Flatten (* Jean-François Alcover, Dec 18 2013, translated from Maple *)
rows = 10;
t = Table[If[n<2, 1, n!/2], {n, 0, rows}];
T[n_, k_] := BellY[n, k, t];
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PROG
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(Sage) # uses[bell_matrix from A264428]
bell_matrix(lambda n: factorial(n)//2 if n>=2 else 1, 8)
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CROSSREFS
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Columns k=0-10 give: A000007, A001710(n-1) for n>0, A215772, A215763, A215764, A215765, A215766, A215767, A215768, A215769, A215770.
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KEYWORD
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AUTHOR
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STATUS
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approved
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