A381682 Triangle read by rows: T(n,k) = number of collections of up to k+1 disjoint subsets of [n] covering [n], with [0]={}, 0<=k<=n.

1, 1, 2, 1, 3, 4, 1, 5, 9, 10, 1, 9, 22, 29, 30, 1, 17, 57, 92, 103, 104, 1, 33, 154, 309, 389, 405, 406, 1, 65, 429, 1080, 1570, 1731, 1753, 1754, 1, 129, 1222, 3889, 6640, 7956, 8250, 8279, 8280, 1, 257, 3537, 14332, 29053, 38650, 41758, 42256, 42293, 42294

Offset

0,3

Comments

Partial row sums of A256894.

For disjoint covers (collections without an empty set) see A102661.

For non-disjoint collections see A381683.

For non-disjoint covers see A369950.

Links

Table of n, a(n) for n=0..54.

Formula

T(n,k) = 2*Sum_{j=1..k} S2(n,j) + S2(n,k+1) for n>=1.

T(0,k) = 1.

Example

Triangle begins:
 1
 1   2
 1   3    4
 1   5    9    10
 1   9   22    29    30
 1  17   57    92   103   104
 1  33  154   309   389   405   406
 1  65  429  1080  1570  1731  1753  1754
 1 129 1222  3889  6640  7956  8250  8279  8280
 1 257 3537 14332 29053 38650 41758 42256 42293 42294
 ...
T(3,2)=9 is the number of disjoint [3]-covering collections of up to 3 subsets:
 {{1,2,3}}
 {{1,2,3},{}}
 {{1},{2,3}}
 {{2},{1,3}}
 {{3},{1,2}}
 {{1},{2},{3}}
 {{1},{2,3},{}}
 {{2},{1,3},{}}
 {{3},{1,2},{}}.

Mathematica

Table[If[n==0, 1, 2*Sum[StirlingS2[n, j], {j, k}] + StirlingS2[n, k+1]], {n, 0, 9}, {k, 0, n}] // Flatten

Crossrefs

Cf. A186021 (diagonal).

Cf. A102661, A256894, A369950, A381683.

Sequence in context: A126198 A055888 A094442 * A060642 A306186 A154929

Adjacent sequences: A381679 A381680 A381681 * A381683 A381684 A381685

Keyword

nonn,tabl

Author

Manfred Boergens, Mar 04 2025

Status

approved