A223894 Triangular array read by rows: T(n,k) is the number of connected components with size k summed over all simple labeled graphs on n nodes; n>=1, 1<=k<=n.
1, 2, 1, 6, 3, 4, 32, 12, 16, 38, 320, 80, 80, 190, 728, 6144, 960, 640, 1140, 4368, 26704, 229376, 21504, 8960, 10640, 30576, 186928, 1866256, 16777216, 917504, 229376, 170240, 326144, 1495424, 14930048, 251548592, 2415919104, 75497472, 11010048, 4902912, 5870592, 17945088, 134370432, 2263937328, 66296291072
Offset: 1
Examples
Triangle T(n,k) begins:
1;
2, 1;
6, 3, 4;
32, 12, 16, 38;
320, 80, 80, 190, 728;
6144, 960, 640, 1140, 4368, 26704;
229376, 21504, 8960, 10640, 30576, 186928, 1866256;
...
Links
- Alois P. Heinz, Rows n = 1..45, flattened
Programs
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Magma
function b(n) // b = A001187 if n eq 0 then return 1; else return 2^Binomial(n,2) - (&+[Binomial(n-1,j-1)*2^Binomial(n-j,2)*b(j): j in [0..n-1]]); end if; return b; end function; A223894:= func< n,k | Binomial(n,k)*2^Binomial(n-k,2)*b(k) >; [A223894(n,k): k in [1..n], n in [1..12]]; // G. C. Greubel, Oct 03 2022
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Maple
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)*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 -
Mathematica
nn = 9; f[list_] := Select[list, # > 0 &]; g = Sum[2^Binomial[n, 2] x^n/n!, {n, 0, nn}]; a = Drop[Range[0, nn]! CoefficientList[Series[Log[g] + 1, {x, 0, nn}], x], 1]; Map[f, Drop[Transpose[Table[Range[0, nn]! CoefficientList[Series[a[[n]] x^n/n! g, {x, 0, nn}], x], {n, 1, nn}]], 1]] // Grid -
SageMath
@CachedFunction def b(n): # b = A001187 if (n==0): return 1 else: return 2^binomial(n,2) - sum(binomial(n-1,j-1)*2^binomial(n-j,2)*b(j) for j in range(n)) def A223894(n,k): return binomial(n,k)*2^binomial(n-k,2)*b(k) flatten([[A223894(n,k) for k in range(1,n+1)] for n in range(1,13)]) # G. C. Greubel, Oct 03 2022
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