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

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

Offset: 3

Author

Vladeta Jovovic, Mar 20 2001

Keywords

• nonn
• tabf

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

Examples

			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).
		

Links

• Andrew Howroyd, Table of n, a(n) for n = 3..1522 (rows n=3..40)

Crossrefs

Row sums are A060491.

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

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

Programs

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

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!.