A228315
Triangular array read by rows: T(n,k) is the number of rooted labeled simple graphs on {1,2,...,n} such that the root is in a component of size k; n>=1, 1<=k<=n.
Original entry on oeis.org
1, 2, 2, 6, 6, 12, 32, 24, 48, 152, 320, 160, 240, 760, 3640, 6144, 1920, 1920, 4560, 21840, 160224, 229376, 43008, 26880, 42560, 152880, 1121568, 13063792, 16777216, 1835008, 688128, 680960, 1630720, 8972544, 104510336, 2012388736
Offset: 1
1;
2, 2;
6, 6, 12;
32, 24, 48, 152;
320, 160, 240, 760, 3640;
- F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 7.
-
b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
T:= (n, k)-> binomial(n, k)*k*b(k)*2^((n-k)*(n-k-1)/2):
seq(seq(T(n, k), k=1..n), n=1..10); # Alois P. Heinz, Aug 26 2013
-
nn = 10; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a =
Drop[Range[0, nn]! CoefficientList[Series[Log[g], {x, 0, nn}], x],
1]; Table[
Table[Binomial[n, k] k a[[k]] 2^Binomial[n - k, 2], {k, 1, n}], {n,
1, 7}] // Grid
A237195
Number of simple labeled graphs on n nodes that contain some size k connected component, all of whose nodes are labeled with integers {1,2,...,k} for some k in {1,2,...,n}.
Original entry on oeis.org
1, 2, 7, 52, 846, 28628, 1928768, 255610528, 66822534992, 34632302913632, 35711543058158592, 73426371674544520192, 301419451958411673103360, 2472252535617096234970201088, 40532629372281642451697543062528, 1328660058258732602631909956943781888
Offset: 1
a(3) = 7. We count all 8 simple labeled graphs on {1,2,3} except: 1-3 2.
-
b:= proc(n) option remember; `if`(n=0, 1, 2^(n*(n-1)/2)-
add(k*binomial(n, k)* 2^((n-k)*(n-k-1)/2)*b(k), k=1..n-1)/n)
end:
a:= n-> add(b(k)*2^((n-k)*(n-k-1)/2), k=1..n):
seq(a(n), n=1..20); # Alois P. Heinz, Feb 04 2014
-
nn=15;g=Sum[2^Binomial[n,2]x^n/n!,{n,0,nn}];a=Drop[Range[0,nn]!CoefficientList[Series[Log[g],{x,0,nn}],x],1];Map[Total,Table[Table[Drop[Transpose[Table[ Range[0,nn]!CoefficientList[Series[a[[n]]x^n/n! g,{x,0,nn}],x],{n,1,nn}]],1][[i,j]]/Binomial[i,j],{j,1,i}],{i,1,nn}]]
A224065
Triangular array read by rows. T(n,k) is the number of size k connected components over all simple unlabeled graphs with n nodes; n>=1,1<=k<=n.
Original entry on oeis.org
1, 2, 1, 4, 1, 2, 8, 3, 2, 6, 19, 5, 4, 6, 21, 53, 14, 10, 12, 21, 112, 209, 39, 24, 24, 42, 112, 853, 1253, 170, 72, 72, 84, 224, 853, 11117, 13599, 1083, 322, 210, 231, 448, 1706, 11117, 261080, 288267, 12516, 2112, 948, 735, 1232, 3412, 22234, 261080, 11716571
Offset: 1
1,
2, 1,
4, 1, 2,
8, 3, 2, 6,
19, 5, 4, 6, 21,
53, 14, 10, 12, 21, 112,
209, 39, 24, 24, 42, 112, 853,
1253, 170, 72, 72, 84, 224, 853, 11117,
13599, 1083, 322, 210, 231, 448, 1706, 11117, 261080,
-
nn=10;h[list_]:=Select[list,#>0&];f[list_]:=Total[Table[list[[i]]*(i-1),{i,1,Length[list]}]];g[x_]:=Sum[NumberOfGraphs[n]x^n,{n,0,nn}];c[x_]:=Sum[a[n]x^n,{n,0,nn}];a[0]=1;sol=SolveAlways[g[x]==Normal[Series[Product[1/(1-x^i)^a[i],{i,1,nn}],{x,0,nn}]],x];b=Drop[Flatten[Table[a[n],{n,0,nn}]/.sol],1];Map[h,Drop[Transpose[Table[Map[f,CoefficientList[Series[(1/(1-y x^n)^b[[n]])Product[1/(1- x^i)^b[[i]],{i,1,nn}](1-x^n)^b[[n]],{x,0,nn}],{x,y}]],{n,1,nn}]],1]]//Flatten
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