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A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).

%I A125205 #13 Feb 27 2022 13:15:35
%S A125205 1,2,1,3,6,3,1,4,18,30,24,15,6,1,5,40,135,250,295,282,215,120,45,10,1,
%T A125205 6,75,420,1385,3015,4800,6365,7170,6705,5065,3009,1365,455,105,15,1,7,
%U A125205 126,1050,5355,18690,47880,96796,166890,251370,329945,373947,362292,297115
%N A125205 Irregular triangle read by rows T(n,k) (n>=1, 0<=k<=n(n-1)/2) giving the total number of connected components in all subgraphs (V,E') with |E'|=k of the complete labeled graph K_n=(V,E).
%F A125205 G.f.: Sum_{n,k} T(n,k)*x^n/n!*y^k=(F(x,y)-1)*exp(F(x,y)-1)=G(x,y)*log(G(x,y)) where G(x,y)=Sum_{n=0..oo} (1+y)^(n(n-1)/2)*x^n/n! and F(x,y)=1+log(G(x,y)) is g.f. of A062734.
%e A125205 Triangle begins:
%e A125205   1;
%e A125205   2,  1;
%e A125205   3,  6,   3,   1;
%e A125205   4, 18,  30,  24,  15,   6,   1;
%e A125205   5, 40, 135, 250, 295, 282, 215, 120, 45, 10, 1;
%e A125205   ...
%e A125205 T(3,1) = 6 since there are three different subgraphs of K_3 with one edge and each subgraph has two connected components.
%o A125205 (PARI) { G=sum(n=0,6,(1+y)^(n*(n-1)/2)*x^n/n!); K=G*log(G); for(n=1,6,print(Vecrev(n!*polcoeff(K,n,x)))) }
%Y A125205 Cf. A062734.
%Y A125205 Cf. A125206 (row-reversed version), A125207 (row sums).
%K A125205 nonn,tabf
%O A125205 1,2
%A A125205 _Max Alekseyev_, Nov 23 2006