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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).

%I A361456 #28 Mar 16 2023 04:50:40
%S A361456 1,1,3,2,13,30,24,6,75,372,780,872,546,180,24,541,4660,18180,42140,
%T A361456 64150,66900,48320,23820,7650,1440,120,4683,62130,385980,1487520,
%U A361456 3973770,7789032,11565360,13238520,11771130,8124710,4314420,1729440,506010,101880,12600,720
%N 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).
%C A361456 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.
%H A361456 E. de Panafieu and S. Dovgal, <a href="https://arxiv.org/abs/1903.09454">Symbolic method and directed graph enumeration</a>, arXiv:1903.09454 [math.CO], 2019.
%F A361456 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!.
%e A361456 Triangle begins:
%e A361456    1;
%e A361456    1;
%e A361456    3,   2;
%e A361456   13,  30,  24,   6;
%e A361456   75, 372, 780, 872, 546, 180, 24;
%e A361456   ...
%t A361456 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]]
%Y A361456 Cf. A334282 (row sums), A000670 (column k=0), A000142 (main diagonal), A046860.
%K A361456 nonn,tabf
%O A361456 0,3
%A A361456 _Geoffrey Critzer_, Mar 12 2023