A361456 Irregular triangle read by rows. T(n,k) is the number of properly colored simple labeled graphs on [n] with exactly k edges, n >= 0, 0 <= k <= binomial(n,2).

1, 1, 3, 2, 13, 30, 24, 6, 75, 372, 780, 872, 546, 180, 24, 541, 4660, 18180, 42140, 64150, 66900, 48320, 23820, 7650, 1440, 120, 4683, 62130, 385980, 1487520, 3973770, 7789032, 11565360, 13238520, 11771130, 8124710, 4314420, 1729440, 506010, 101880, 12600, 720

Offset

0,3

Comments

The graphs of order n are properly colored from the color set {c_1, c_2,...,c_n} such that if c_i is used as a color then c_{i-1} is also used as a color.

Links

Table of n, a(n) for n=0..41.

E. de Panafieu and S. Dovgal, Symbolic method and directed graph enumeration, arXiv:1903.09454 [math.CO], 2019.

Formula

Sum_{n>=0} Sum_{k>=0} T(n,k)*w^k*z^n/((1+w)^binomial(n,2)*n!) = 1/(1-(E(z,w)-1)) where E(z,w) = Sum_{n>=0} z^n/(1+w)^binomial(n,2)*n!.

Example

Triangle begins:
   1;
   1;
   3,   2;
  13,  30,  24,   6;
  75, 372, 780, 872, 546, 180, 24;
  ...

Mathematica

nn = 8; e[z_, w_] := Sum[z^n/(n! (1 + w)^Binomial[n, 2]), {n, 0, Binomial[nn, 2]}]; Map[CoefficientList[Series[#, {w, 0, Binomial[nn, 2]}], w] &, Table[n! (1 + w)^Binomial[n, 2], {n, 0, nn}] CoefficientList[Series[1/(1 - (e[z, w] - 1)), {z, 0, nn}], z]]

Crossrefs

Cf. A334282 (row sums), A000670 (column k=0), A000142 (main diagonal), A046860.

Sequence in context: A098384 A243253 A064536 * A324012 A355259 A231183

Adjacent sequences: A361453 A361454 A361455 * A361457 A361458 A361459

Keyword

nonn,tabf

Author

Geoffrey Critzer, Mar 12 2023

Status

approved