A331123 - OEIS

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.

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:

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-François Alcover in A001349 *)

CROSSREFS

Cf. A217955.

Sequence in context: A147865 A266614 A175314 * A182223 A011152 A078298

Adjacent sequences: A331120 A331121 A331122 * A331124 A331125 A331126

KEYWORD

nonn,tabf

AUTHOR

Geoffrey Critzer, Jan 10 2020

STATUS

approved