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A060487 Triangle T(n,k) of k-block tricoverings of an n-set (n >= 3, k >= 4). 11
1, 3, 1, 7, 57, 95, 43, 3, 35, 717, 3107, 4520, 2465, 445, 12, 155, 7845, 75835, 244035, 325890, 195215, 50825, 4710, 70, 651, 81333, 1653771, 10418070, 27074575, 33453959, 20891962, 6580070, 965965, 52430, 465 (list; graph; refs; listen; history; text; internal format)
OFFSET

3,2

COMMENTS

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

LINKS

Andrew Howroyd, Table of n, a(n) for n = 3..1157

FORMULA

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).

EXAMPLE

Triangle begins:

  [1, 3, 1];

  [7, 57, 95, 43, 3];

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

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

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

   ...

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.

PROG

(PARI)

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

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]!)}

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))}

for(n=3, 8, print(Vecrev(row(3, n)))); \\ Andrew Howroyd, Dec 23 2018

CROSSREFS

Columns include A060483, A060484, A060485.

Row sums are A060486.

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

Sequence in context: A282422 A282685 A194583 * A285020 A165781 A152095

Adjacent sequences:  A060484 A060485 A060486 * A060488 A060489 A060490

KEYWORD

nonn,tabf

AUTHOR

Vladeta Jovovic, Mar 20 2001

STATUS

approved

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Last modified July 25 23:39 EDT 2021. Contains 346294 sequences. (Running on oeis4.)