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.

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.

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