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A215771 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.

%I A215771 #38 Mar 23 2020 12:10:46
%S A215771 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,
%T A215771 360,777,742,420,140,21,1,0,2520,5547,5446,3157,1190,266,28,1,0,20160,
%U A215771 45216,45559,27342,10857,2898,462,36,1,0,181440,414144,427275,264925,109935,31899,6300,750,45,1
%N A215771 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.
%C A215771 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
%H A215771 Alois P. Heinz, <a href="/A215771/b215771.txt">Rows n = 0..140, flattened</a>
%e A215771 T(4,1) = 3:  .1-2.  .1 2.  .1-2.
%e A215771 .            .| |.  .|X|.  . X .
%e A215771 .            .3-4.  .3 4.  .3-4.
%e A215771 .
%e A215771 T(4,2) = 7:  .1 2.  .1-2.  .1 2.  o1 2.  .1 2o  .1-2.  .1-2.
%e A215771 .            .| |.  .   .  . X .  . /|.  .|\ .  . \|.  .|/ .
%e A215771 .            .3 4.  .3-4.  .3 4.  .3-4.  .3-4.  o3 4.  .3 4o
%e A215771 .
%e A215771 T(4,3) = 6:  .1 2o  .1-2.  o1 2.  o1 2o  o1 2.  .1 2o
%e A215771 .            .|  .  .   .  .  |.  .   .  . / .  . \ .
%e A215771 .            .3 4o  o3 4o  o3 4.  .3-4.  .3 4o  o3 4.
%e A215771 .
%e A215771 T(4,4) = 1:  o1 2o
%e A215771 .            .   .
%e A215771 .            o3 4o
%e A215771 Triangle T(n,k) begins:
%e A215771   1;
%e A215771   0,   1;
%e A215771   0,   1,   1;
%e A215771   0,   1,   3,   1;
%e A215771   0,   3,   7,   6,   1;
%e A215771   0,  12,  25,  25,  10,   1;
%e A215771   0,  60, 127, 120,  65,  15,  1;
%e A215771   0, 360, 777, 742, 420, 140, 21,  1;
%p A215771 T:= proc(n, k) option remember; `if`(k<0 or k>n, 0, `if`(n=0, 1,
%p A215771       add(binomial(n-1, i)*T(n-1-i, k-1)*ceil(i!/2), i=0..n-k)))
%p A215771     end:
%p A215771 seq(seq(T(n, k), k=0..n), n=0..12);
%p A215771 # Alternatively, with the function BellMatrix defined in A264428:
%p A215771 BellMatrix(n -> `if`(n<2, 1, n!/2), 8); # _Peter Luschny_, Jan 21 2016
%t A215771 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 *)
%t A215771 rows = 10;
%t A215771 t = Table[If[n<2, 1, n!/2], {n, 0, rows}];
%t A215771 T[n_, k_] := BellY[n, k, t];
%t A215771 Table[T[n, k], {n, 0, rows}, {k, 0, n}] // Flatten (* _Jean-François Alcover_, Jun 22 2018, after _Peter Luschny_ *)
%o A215771 (Sage) # uses[bell_matrix from A264428]
%o A215771 bell_matrix(lambda n: factorial(n)//2 if n>=2 else 1, 8)
%Y A215771 Columns k=0-10 give: A000007, A001710(n-1) for n>0, A215772, A215763, A215764, A215765, A215766, A215767, A215768, A215769, A215770.
%Y A215771 Diagonal and lower diagonals give: A000012, A000217, A001296, A215773, A215774.
%Y A215771 Row sums give A002135.
%Y A215771 T(2n,n) gives A253276.
%K A215771 nonn,tabl
%O A215771 0,9
%A A215771 _Alois P. Heinz_, Aug 23 2012