A201922 Triangle read by rows: T(n,m) = number of unlabeled graphs on n nodes with m connected components, m = 1,2,...,n.

1, 1, 1, 2, 1, 1, 6, 3, 1, 1, 21, 8, 3, 1, 1, 112, 30, 9, 3, 1, 1, 853, 145, 32, 9, 3, 1, 1, 11117, 1028, 154, 33, 9, 3, 1, 1, 261080, 12320, 1065, 156, 33, 9, 3, 1, 1, 11716571, 274806, 12513, 1074, 157, 33, 9, 3, 1, 1, 1006700565, 12007355, 276114, 12550, 1076, 157, 33, 9, 3, 1, 1

Offset

1,4

Links

Alois P. Heinz, Rows n = 1..75, flattened

P. Flajolet, R. Sedgewick, Analytic combinatorics, Theorem I.1 (Multiset)

R. J. Mathar, Statistics on Small Graphs, arXiv:1709.09000 (2017) Table 82

Peter Steinbach, Field Guide to Simple Graphs, Volume 4, Part 5 (For Volumes 1, 2, 3, 4 of this book see A000088, A008406, A000055, A000664, respectively.)

Formula

T(n,m) = sum over the partitions of n with m parts: 1*K1 + 2*K2 + ... + n*Kn = n, K1 + K2 + ... + Kn = m, of Product_{i=1..n} binomial(A001349(i) + Ki - 1, Ki).

O.g.f.: Product_{n>=1} 1/(1 - y*x^n)^A001349(n). - Geoffrey Critzer, Apr 19 2012

Example

Triangle starts:
    1
    1   1
    2   1   1
    6   3   1   1
   21   8   3   1   1
  112  30   9   3   1   1
  853 145  32   9   3   1   1 ...

Mathematica

nn=10; c=(A000088=Table[NumberOfGraphs[n], {n, 0, nn}]; f[x_] = 1-Product[1/(1-x^k)^a[k], {k, 1, nn}]; a[0]=a[1]=a[2]=1; coes=CoefficientList[Series[f[x], {x, 0, nn}], x]; sol=First[Solve[Thread[Rest[coes+A000088]==0]]]; Table[a[n], {n, 0, nn}]/.sol); f[list_]:=Select[list, #>0&]; g=Product[1/(1-y x^n)^c[[n+1]], {n, 1, nn}]; Map[f, Drop[CoefficientList[Series[g, {x, 0, nn}], {x, y}], 1]] //Flatten (* Geoffrey Critzer, Apr 19 2012  (c in above Mma code is given by Jean Francois Alcover in A001349) *)

Crossrefs

Cf. A001349 (first column), A000088 (row sum), A201968 (limits in the diagonals), A106240, A274934 (2nd column).

Sequence in context: A201198 A349933 A120258 * A181644 A144351 A213936

Adjacent sequences: A201919 A201920 A201921 * A201923 A201924 A201925

Keyword

nonn,tabl,nice

Author

Max Alekseyev, Dec 06 2011

Status

approved