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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 (list; table; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS Row sums are A224031. Column 1 is A006897. T(n,n) is A001349. LINKS Table of n, a(n) for n=1..55. 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 Adjacent sequences: A224062 A224063 A224064 * A224066 A224067 A224068 KEYWORD nonn,tabl AUTHOR Geoffrey Critzer, Mar 30 2013 STATUS approved

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Last modified August 7 17:47 EDT 2024. Contains 375017 sequences. (Running on oeis4.)