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A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).

%I A060492 #12 Jan 30 2020 16:35:25
%S A060492 4,60,120,13,375,3030,9030,5040,28,1392,24552,207900,838320,1345680,
%T A060492 362880,50,4020,130740,2208430,20334720,101752560,257065200,261122400,
%U A060492 46569600,80,9960,551640,16365410,274814760,2709457128,15812198640
%N A060492 Triangle T(n,k) of k-block ordered tricoverings of an unlabeled n-set (n >= 3, k = 4..2n).
%C A060492 A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
%C A060492 All columns are polynomials of order binomial(k, 3). - _Andrew Howroyd_, Jan 30 2020
%H A060492 Andrew Howroyd, <a href="/A060492/b060492.txt">Table of n, a(n) for n = 3..1522</a> (rows n=3..40)
%F A060492 E.g.f. for ordered k-block tricoverings of an unlabeled n-set is exp(-x+x^2/2+x^3/3*y/(1-y))*Sum_{k=0..inf}1/(1-y)^binomial(k, 3)*exp(-x^2/2*1/(1-y)^n)*x^k/k!.
%e A060492 Triangle begins:
%e A060492   [4, 60, 120],
%e A060492   [13, 375, 3030, 9030, 5040],
%e A060492   [28, 1392, 24552, 207900, 838320, 1345680, 362880],
%e A060492   [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
%e A060492    ...
%e A060492 There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).
%o A060492 (PARI) \\ gives g.f. of k-th column.
%o A060492 ColGf(k) = k!*polcoef(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^k) )*sum(j=0, k, 1/(1-y)^binomial(j, 3)*exp((-x^2/2)/(1-y)^j + O(x*x^k))*x^j/j!), k) \\ _Andrew Howroyd_, Jan 30 2020
%o A060492 (PARI)
%o A060492 T(n)={my(m=2*n, y='y + O('y^(n+1))); my(g=serlaplace(exp(-x + x^2/2 + x^3*y/(3*(1-y)) + O(x*x^m))*sum(k=0, m, 1/(1-y)^binomial(k, 3)*exp((-x^2/2)/(1-y)^k + O(x*x^m))*x^k/k!))); Mat([Col(p/y^3, -n) | p<-Vec(g)[2..m+1]])}
%o A060492 { my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ _Andrew Howroyd_, Jan 30 2020
%Y A060492 Row sums are A060491.
%Y A060492 Columns k=4..6 are A060488, A060489, A060490.
%Y A060492 Cf. A059443, A059530, A060052, A060092, A060487, A331571.
%K A060492 nonn,tabf
%O A060492 3,1
%A A060492 _Vladeta Jovovic_, Mar 20 2001