%I #10 Dec 23 2018 16:14:39

%S 1,3,1,7,57,95,43,3,35,717,3107,4520,2465,445,12,155,7845,75835,

%T 244035,325890,195215,50825,4710,70,651,81333,1653771,10418070,

%U 27074575,33453959,20891962,6580070,965965,52430,465

%N Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4).

%C A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.

%H Andrew Howroyd, <a href="/A060487/b060487.txt">Table of n, a(n) for n = 3..1157</a>

%F E.g.f. for k-block tricoverings of an n-set is exp(-x+x^2/2+(exp(y)-1)*x^3/3)*Sum_{k=0..inf}x^k/k!*exp(-1/2*x^2*exp(k*y))*exp(binomial(k, 3)*y).

%e Triangle begins:

%e [1, 3, 1];

%e [7, 57, 95, 43, 3];

%e [35, 717, 3107, 4520, 2465, 445, 12];

%e [155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70];

%e [651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465];

%e ...

%e There are 205 tricoverings of a 4-set(cf. A060486): 7 4-block, 57 5-block, 95 6-block, 43 7-block and 3 8-block tricoverings.

%o (PARI)

%o WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}

%o D(p, n, k)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); WeighT(v)[n]^k/prod(i=1, #v, i^v[i]*v[i]!)}

%o row(n, k)={my(m=n*k+1, q=Vec(exp(intformal(O(x^m) - x^n/(1-x)))/(y+x))); if(n==0, 1, (-1)^m*sum(j=0, m, my(s=0); forpart(p=j, s+=(-1)^#p*D(p, n, k), [1, n]); s*q[#q-j])*y^(m-n)/(1+y))}

%o for(n=3, 8, print(Vecrev(row(3,n)))); \\ _Andrew Howroyd_, Dec 23 2018

%Y Columns include A060483, A060484, A060485.

%Y Row sums are A060486.

%Y Cf. A006095, A060090-A060095, A060069, A060070, A060051-A060053, A002718, A059443, A003462, A059945-A059951.

%K nonn,tabf

%O 3,2

%A _Vladeta Jovovic_, Mar 20 2001