A094573 Triangle T(n,k) giving number of (<=2)-covers of an n-set with k blocks.

1, 1, 1, 3, 1, 1, 12, 20, 7, 1, 39, 169, 186, 59, 3, 1, 120, 1160, 2755, 2243, 661, 55, 1, 363, 7381, 33270, 52060, 33604, 9167, 910, 15, 1, 1092, 45500, 367087, 988750, 1126874, 601262, 151726, 16401, 525, 1, 3279, 276529, 3873786, 17005149

Offset: 0

Author

Goran Kilibarda, Vladeta Jovovic, May 12 2004

Keywords

• nonn
• tabf

Comments

Cover of a set is (<=2)-cover if every element of the set is covered with at most two blocks of the cover.

Examples

			Triangle T(n,k) begins:
  1;
  1;
  1,   3,    1;
  1,  12,   20,    7;
  1,  39,  169,  186,   59,   3;
  1, 120, 1160, 2755, 2243, 661, 55;
  ...
		

Crossrefs

Row sums give A094574.

Cf. A059443, A060052.

Programs

Mathematica

rows = 9; m = rows + 2;
egf = Exp[-x - (x^2/2)*(Exp[y]-1)]*Sum[Exp[y*Binomial[n+1, 2]]*(x^n/n!), {n, 0, m}];
cc = CoefficientList[# + O[x]^m, x]& /@ CoefficientList[egf + O[y]^m, y];
(Range[0, Length[cc]-1]! * cc)[[1 ;; rows]] /. {0, a__} :> {a} // Flatten (* Jean-François Alcover, May 13 2019 *)

Formula

E.g.f.: exp(-x-x^2/2*(exp(y)-1))*(Sum_{n>=0} exp(y*binomial(n+1, 2))*x^n/n!).