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A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.

%I A059530 #16 Jan 31 2020 14:31:38
%S A059530 0,0,0,0,1,3,1,0,0,0,0,1,39,89,43,3,0,0,0,0,0,252,2192,4090,2435,445,
%T A059530 12,0,0,0,0,0,1260,37080,179890,289170,188540,50645,4710,70,0,0,0,0,0,
%U A059530 5040,536760,6052730,20660055,29432319,19826737,6481160,964495,52430
%N A059530 Triangle T(n,k) of k-block T_0-tricoverings of an n-set, n >= 3, k = 0..2*n.
%C A059530 A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering. A covering of a set is a T_0-covering if for every two distinct elements of the set there exists a block of the covering containing one but not the other element.
%D A059530 I. P. Goulden and D. M. Jackson, Combinatorial Enumeration, John Wiley and Sons, N.Y., 1983.
%H A059530 Andrew Howroyd, <a href="/A059530/b059530.txt">Table of n, a(n) for n = 3..1674</a>
%H A059530 Vladeta Jovovic, <a href="/A060070/a060070.pdf">T_0-tricoverings of a 4-set</a>
%F A059530 E.g.f. for k-block T_0-tricoverings of an n-set is exp(-x+1/2*x^2+1/3*x^3*y)*Sum_{i=0..inf}(1+y)^binomial(i, 3)*exp(-1/2*x^2*(1+y)^i)*x^i/i!.
%F A059530 T(n,k) = 0 for n > binomial(k, 3). - _Andrew Howroyd_, Jan 30 2020
%e A059530 Triangle begins:
%e A059530   [0, 0, 0, 0, 1, 3, 1],
%e A059530   [0, 0, 0, 0, 1, 39, 89, 43, 3],
%e A059530   [0, 0, 0, 0, 0, 252, 2192, 4090, 2435, 445, 12],
%e A059530   [0, 0, 0, 0, 0, 1260, 37080, 179890, 289170, 188540, 50645, 4710, 70],
%e A059530    ...
%e A059530 There are 5 = 1 + 3 + 1 T_0-tricoverings of a 3-set and 175 = 1 + 39 + 89 + 43 + 3 T_0-tricoverings of a 4-set, cf. A060070.
%o A059530 (PARI) \\ gets k-th column as vector
%o A059530 C(k)=if(k<4, [], Vecrev(serlaplace(polcoef(exp(-x + x^2/2 + x^3*y/3 + O(x*x^k))*sum(i=0, 2*k, (1+y)^binomial(i, 3)*exp(-x^2*(1+y)^i/2 + O(x*x^k))*x^i/i!), k))/y)) \\ _Andrew Howroyd_, Jan 30 2020
%o A059530 (PARI)
%o A059530 T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=exp(-x + x^2/2 + x^3*y/3 + O(x*x^m))*sum(k=0, m, (1+y)^binomial(k, 3)*exp(-x^2*(1+y)^k/2 + O(x*x^m))*x^k/k!)); Mat([Col(serlaplace(p), -n) | p<-Vec(g)[2..m+1]]);}
%o A059530 { my(A=T(8)); for(n=3, matsize(A)[1], print(concat([0], A[n, 1..2*n]))) } \\ _Andrew Howroyd_, Jan 30 2020
%Y A059530 Column sums are A060069.
%Y A059530 Row sums are A060070.
%Y A059530 Cf. A059443, A060052, A060092, A060487, A060492, A331039.
%K A059530 nonn,tabf
%O A059530 3,6
%A A059530 _Vladeta Jovovic_, Feb 22 2001