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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.
0
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
1,2
COMMENTS
Row sums are A224031.
Column 1 is A006897.
T(n,n) is A001349.
FORMULA
O.g.f. for column k is the derivative with respect to y then evaluated at y = 1 of (1/(1 - y*x^k))^A001349(k) * (1 - x^k)^A001349(k) * Product_{k>=1}1/(1 - x^k)^A001349(k).
EXAMPLE
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,
MATHEMATICA
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
CROSSREFS
Cf. A223894 (labeled version).
Sequence in context: A106616 A268669 A030652 * A077904 A088964 A326721
KEYWORD
nonn,tabl
AUTHOR
Geoffrey Critzer, Mar 30 2013
STATUS
approved