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A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.

%I A322278 #16 Dec 05 2018 16:01:44
%S A322278 1,0,1,0,3,4,0,19,84,38,0,195,2470,3140,728,0,3031,108390,307390,
%T A322278 186360,26704,0,67263,7192444,42747460,52630060,18926544,1866256,0,
%U A322278 2086099,726782784,9030799218,20784069600,14401134944,3463311488,251548592
%N A322278 Triangle read by rows: T(n,k) is the number of k-colored connected graphs on n labeled nodes up to permutation of the colors.
%C A322278 Equivalently, the number of ways to choose a stable partition of a simple connected graph on n labeled nodes with k parts. See A322064 for the definition of stable partition.
%H A322278 Andrew Howroyd, <a href="/A322278/b322278.txt">Table of n, a(n) for n = 1..1275</a>
%F A322278 T(n,k) = (1/k!)*Sum_{j=0..k} (-1)^(k-j)*binomial(k,j)*A322279(n,j).
%e A322278 Triangle begins:
%e A322278   1;
%e A322278   0,     1;
%e A322278   0,     3,       4;
%e A322278   0,    19,      84,       38;
%e A322278   0,   195,    2470,     3140,      728;
%e A322278   0,  3031,  108390,   307390,   186360,    26704;
%e A322278   0, 67263, 7192444, 42747460, 52630060, 18926544, 1866256;
%e A322278   ...
%o A322278 (PARI)
%o A322278 M(n, K=n)={
%o A322278   my(p=sum(j=0, n, x^j/(j!*2^binomial(j, 2))) + O(x*x^n));
%o A322278   my(q=sum(j=0, n, x^j*2^binomial(j, 2)) + O(x*x^n));
%o A322278   my(W=vector(K, k, Col(serlaplace(log(serconvol(q, p^k))))));
%o A322278   Mat(vector(K, k, sum(i=1, k, (-1)^(k-i)*binomial(k,i)*W[i])/k!));
%o A322278 }
%o A322278 my(T=M(7)); for(n=1, #T, print(T[n, 1..n]))
%Y A322278 Row sums are A322064.
%Y A322278 Columns k=2..4 are A001832(for n > 1), A322330, A322331.
%Y A322278 Right diagonal is A001187.
%Y A322278 Cf. A058843, A058875, A322279, A322280.
%K A322278 nonn,tabl
%O A322278 1,5
%A A322278 _Andrew Howroyd_, Dec 01 2018