OFFSET
3,1
COMMENTS
A covering of a set is a tricovering if every element of the set is covered by exactly three blocks of the covering.
All columns are polynomials of order binomial(k, 3). - Andrew Howroyd, Jan 30 2020
LINKS
Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)
FORMULA
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!.
EXAMPLE
Triangle begins:
[4, 60, 120],
[13, 375, 3030, 9030, 5040],
[28, 1392, 24552, 207900, 838320, 1345680, 362880],
[50, 4020, 130740, 2208430, 20334720, 101752560, 257065200, 261122400, 46569600], [80, 9960, 551640, 16365410, 274814760, 2709457128, 15812198640, 52897521600, 91945022400, 64778313600, 8043235200],
...
There are 184 ordered tricoverings of an unlabeled 3-set: 4 4-block, 60 5-block and 120 6-block tricoverings (cf. A060491).
PROG
(PARI) \\ gives g.f. of k-th column.
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
(PARI)
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]])}
{ my(A=T(8)); for(n=3, matsize(A)[1], print(A[n, 4..2*n])) } \\ Andrew Howroyd, Jan 30 2020
CROSSREFS
KEYWORD
nonn,tabf
AUTHOR
Vladeta Jovovic, Mar 20 2001
STATUS
approved