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A331123 Triangular array read by rows.  T(n,k) is the number of simple unlabeled graphs with n vertices whose components belong to exactly k distinct isomorphism classes. 0
1, 2, 3, 1, 8, 3, 22, 12, 116, 38, 2, 854, 181, 9, 11125, 1176, 45, 261083, 13351, 233, 1, 11716594, 287048, 1513, 13, 1006700566, 12281514, 15707, 77, 164059830598, 1031031446, 310050, 498, 50335907869220, 166110813984, 12681157, 3585, 6, 29003487462848916, 50667148763414, 1045586096, 37005, 57, 31397381142761241984, 29104659809235092, 167233146488, 684742, 462 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

LINKS

Table of n, a(n) for n=1..48.

FORMULA

G.f.: Product_{k>=1} (y/(1-x^k) - y + 1)^A001349(k).

EXAMPLE

Triangle begins:

         1;

         2;

         3,      1;

         8,      3;

        22,     12;

       116,     38,    2;

       854,    181,    9;

     11125,   1176,   45;

    261083,  13351,  233,  1;

  11716594, 287048, 1513, 13;

T(4,2)=3 because we have *-* * * , *-*-* * , a triangle with an isolated point.

MATHEMATICA

Needs["Combinatorica`"]; max = 10;

A000088 = Table[NumberOfGraphs[n], {n, 0, max}];

f[x_] = 1 - Product[1/(1 - x^k)^a[k], {k, 1, max}];

a[0] = a[1] = a[2] = 1; coes = CoefficientList[Series[f[x], {x, 0, max}], x];

sol = Solve[Thread[Rest[coes + A000088] == 0]];

c = Drop[Table[a[n], {n, 0, max}] /. sol // Flatten, 1];

Map[Select[#, # > 0 &] &, Drop[CoefficientList[ Series[Product[(y/(1 - x^k) - y + 1)^c[[k]], {k, 1, max}], {x, 0, max}], {x, y}], 1]] // Grid (* after code by Jean-Fran├žois Alcover in A001349 *)

CROSSREFS

Cf. A217955.

Sequence in context: A263322 A266614 A175314 * A182223 A011152 A078298

Adjacent sequences:  A331120 A331121 A331122 * A331124 A331125 A331126

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Jan 10 2020

STATUS

approved

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Last modified July 13 17:54 EDT 2020. Contains 335689 sequences. (Running on oeis4.)