A381683 Triangle read by rows: T(n,k) = number of collections of up to k subsets of [n] covering [n], with [0]={}; n>=0, k=0..2^n.

1, 2, 0, 1, 2, 0, 1, 5, 9, 10, 0, 1, 14, 58, 125, 181, 209, 217, 218, 0, 1, 41, 401, 1947, 6091, 13987, 25395, 38261, 49701, 57709, 62077, 63897, 64457, 64577, 64593, 64594, 0, 1, 122, 2802, 30352, 210448, 1076880, 4385616, 14839576, 42831176, 107303376, 236306016, 462089756, 809460556, 1280895556, 1846618196, 2447698581

Offset: 0

Author

Manfred Boergens, Mar 04 2025

Keywords

• nonn
• tabf

Comments

Partial row sums of A163353.

For covers (collections without an empty set) see A369950.

For disjoint collections see A381682.

For disjoint covers see A102661.

Examples

			Triangle begins:
  1 2
  0 1  2
  0 1  5   9   10
  0 1 14  58  125  181   209   217   218
  0 1 41 401 1947 6091 13987 25395 38261 49701 57709 62077 63897 64457 64577 64593 64594
  ...
T(3,2)=14 is the number of covering collections of 1 or 2 subsets of [3]:
  {{1,2,3}}
  {{},{1,2,3}}
  {{1},{2,3}}
  {{1},{1,2,3}}
  {{2},{1,3}}
  {{2},{1,2,3}}
  {{3},{1,2}}
  {{3},{1,2,3}}
  {{1,2},{1,3}}
  {{1,2},{2,3}}
  {{1,3},{2,3}}
  {{1,2},{1,2,3}}
  {{1,3},{1,2,3}}
  {{2,3},{1,2,3}}.
		

Crossrefs

Cf. A000371 (diagonal).

Cf. A102661, A163353, A369950, A381682.

Programs

Mathematica

Table[Sum[Sum[(-1)^(n-i)*Binomial[n, i]*Binomial[2^i, j], {i, 0, n}], {j, 0, k}], {n, 0, 4}, {k, 0, 2^n}]//Flatten

PARI

T(n,k) = sum(j=0,k, sum(i=0,n, (-1)^(n-i)*binomial(n,i)*binomial(2^i,j)));
for(n=0,5,for(k=0,2^n,print1(T(n,k),", "))); \\ Joerg Arndt, Mar 04 2025

Formula

T(n,k) = Sum_{j=0..k} Sum_{i=0..n} (-1)^(n-i)*binomial(n,i)*binomial(2^i,j).