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A060514 Triangle T(n,k) of series-reduced (or homeomorphically irreducible) labeled graphs with n nodes and k edges, k=0..binomial(n,2).

%I A060514 #11 Jan 23 2020 03:49:36
%S A060514 1,1,1,1,1,3,0,0,1,6,3,4,0,0,1,1,10,15,20,5,0,5,20,15,10,1,1,15,45,75,
%T A060514 90,96,135,315,510,760,843,765,395,105,15,1,1,21,105,245,525,777,1302,
%U A060514 3045,7455,16275,30135,50190,70805,81690,70605,43239,18774,5880,1330
%N A060514 Triangle T(n,k) of series-reduced (or homeomorphically irreducible) labeled graphs with n nodes and k edges, k=0..binomial(n,2).
%D A060514 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H A060514 D. M. Jackson and J. W. Reilly, <a href="https://doi.org/10.1016/0095-8956(75)90090-8">The enumeration of homeomorphically irreducible labeled graphs</a>, J. Combin. Theory, B 19 (1975), 272-286.
%F A060514 E.g.f. : (1+x*y)^(-1/2)*exp(x*y/2-x^2*y^2/4)*Sum_{k=0..inf}((1+x)*exp(-x^2*y/(1+x*y)))^binomial(k, 2)*(exp(1/2*x^3*y^2/(1+x*y)))^k*x^k/k!
%e A060514 Triangle begins:
%e A060514 [1],
%e A060514 [1],
%e A060514 [1, 1],
%e A060514 [1, 3, 0, 0],
%e A060514 [1, 6, 3, 4, 0, 0, 1],
%e A060514 [1, 10, 15, 20, 5, 0, 5, 20, 15, 10, 1],
%e A060514 [1, 15, 45, 75, 90, 96, 135, 315, 510, 760, 843, 765, 395, 105, 15, 1],
%e A060514 [1, 21, 105, 245, 525, 777, 1302, 3045, 7455, 16275, 30135, 50190, 70805, 81690, 70605, 43239, 18774, 5880, 1330, 210, 21, 1],
%e A060514 ...
%Y A060514 Row sums: A003514.
%Y A060514 For connected graphs see A331437, A331438.
%K A060514 nonn,tabf
%O A060514 0,6
%A A060514 _Vladeta Jovovic_, Mar 23 2001