cp's OEIS Frontend

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.

%I A381682 #11 Mar 07 2025 06:35:58
%S A381682 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,
%T A381682 389,405,406,1,65,429,1080,1570,1731,1753,1754,1,129,1222,3889,6640,
%U A381682 7956,8250,8279,8280,1,257,3537,14332,29053,38650,41758,42256,42293,42294
%N 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.
%C A381682 Partial row sums of A256894.
%C A381682 For disjoint covers (collections without an empty set) see A102661.
%C A381682 For non-disjoint collections see A381683.
%C A381682 For non-disjoint covers see A369950.
%F A381682 T(n,k) = 2*Sum_{j=1..k} S2(n,j) + S2(n,k+1) for n>=1.
%F A381682 T(0,k) = 1.
%e A381682 Triangle begins:
%e A381682  1
%e A381682  1   2
%e A381682  1   3    4
%e A381682  1   5    9    10
%e A381682  1   9   22    29    30
%e A381682  1  17   57    92   103   104
%e A381682  1  33  154   309   389   405   406
%e A381682  1  65  429  1080  1570  1731  1753  1754
%e A381682  1 129 1222  3889  6640  7956  8250  8279  8280
%e A381682  1 257 3537 14332 29053 38650 41758 42256 42293 42294
%e A381682  ...
%e A381682 T(3,2)=9 is the number of disjoint [3]-covering collections of up to 3 subsets:
%e A381682  {{1,2,3}}
%e A381682  {{1,2,3},{}}
%e A381682  {{1},{2,3}}
%e A381682  {{2},{1,3}}
%e A381682  {{3},{1,2}}
%e A381682  {{1},{2},{3}}
%e A381682  {{1},{2,3},{}}
%e A381682  {{2},{1,3},{}}
%e A381682  {{3},{1,2},{}}.
%t A381682 Table[If[n==0, 1, 2*Sum[StirlingS2[n, j], {j, k}] + StirlingS2[n, k+1]], {n, 0, 9}, {k, 0, n}] // Flatten
%Y A381682 Cf. A186021 (diagonal).
%Y A381682 Cf. A102661, A256894, A369950, A381683.
%K A381682 nonn,tabl
%O A381682 0,3
%A A381682 _Manfred Boergens_, Mar 04 2025