A060492 - OEIS

%I #12 Jan 30 2020 16:35:25

%S 4,60,120,13,375,3030,9030,5040,28,1392,24552,207900,838320,1345680,

%T 362880,50,4020,130740,2208430,20334720,101752560,257065200,261122400,

%U 46569600,80,9960,551640,16365410,274814760,2709457128,15812198640

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

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

%C All columns are polynomials of order binomial(k, 3). - _Andrew Howroyd_, Jan 30 2020

%H Andrew Howroyd, <a href="/A060492/b060492.txt">Table of n, a(n) for n = 3..1522</a> (rows n=3..40)

%F 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 Triangle begins:

%e   [4, 60, 120],

%e   [13, 375, 3030, 9030, 5040],

%e   [28, 1392, 24552, 207900, 838320, 1345680, 362880],

%e   [50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],

%e    ...

%e There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).

%o (PARI) \\ gives g.f. of k-th column.

%o 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 (PARI)

%o 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 { my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ _Andrew Howroyd_, Jan 30 2020

%Y Row sums are A060491.

%Y Columns k=4..6 are A060488, A060489, A060490.

%Y Cf. A059443, A059530, A060052, A060092, A060487, A331571.

%K nonn,tabf

%O 3,1

%A _Vladeta Jovovic_, Mar 20 2001