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A369950 Triangle read by rows: T(n,k) = number of j-covers of [n] with j<=k, k=1..2^n-1.

%I A369950 #38 Mar 09 2025 12:27:26
%S A369950 1,1,4,5,1,13,45,80,101,108,109,1,40,361,1586,4505,9482,15913,22348,
%T A369950 27353,30356,31721,32176,32281,32296,32297,1,121,2681,27671,182777,
%U A369950 894103,3491513,11348063,31483113,75820263,160485753,301604003
%N A369950 Triangle read by rows: T(n,k) = number of j-covers of [n] with j<=k, k=1..2^n-1.
%C A369950 Partial row sums of A055154.
%C A369950 Also, number of k-covers of [n] allowing for empty subsets.
%C A369950 Amendments by _Manfred Boergens_, Mar 09 2025: (Start)
%C A369950 For covers which may include one empty set see A381683.
%C A369950 For disjoint covers see A102661.
%C A369950 For disjoint covers which may include one empty set see A381682. (End)
%H A369950 John Tyler Rascoe, <a href="/A369950/b369950.txt">Rows n = 1..9, flattened</a>
%F A369950 T(n,k) = Sum_{i=1..k} (1/i!)*Sum_{j=0..i} Stirling1(i+1, j+1)*(2^j-1)^n.
%F A369950 T(n,k) = Sum_{i=1..k} Sum_{j=0..n} (-1)^j*C(n, j)*C(2^(n-j)-1, i).
%F A369950 T(n,2^n-1) = A003465(n).
%e A369950 Triangle (with rows n >= 1 and columns k >= 1) begins as follows:
%e A369950  1;
%e A369950  1, 4, 5;
%e A369950  1, 13, 45, 80, 101, 108, 109;
%e A369950  1, 40, 361, 1586, 4505, 9482, 15913, 22348, 27353, 30356, 31721, 32176, 32281, 32296, 32297;
%e A369950  ...
%e A369950 There are T(3,2) = 13 covers of [3] consisting of up to 2 subsets (brackets and commas omitted):
%e A369950  123
%e A369950  123 1
%e A369950  123 2
%e A369950  123 3
%e A369950  123 12
%e A369950  123 13
%e A369950  123 23
%e A369950  12 13
%e A369950  12 23
%e A369950  13 23
%e A369950  12 3
%e A369950  13 2
%e A369950  23 1
%t A369950 Flatten[Table[Sum[Sum[StirlingS1[i+1, j+1] (2^j-1)^n, {j, 0, i}]/i!, {i, k}], {n, 6}, {k, 2^n-1}]]
%o A369950 (Python)
%o A369950 from math import comb
%o A369950 def A369950(n,k): return sum((-1)**j*comb(n, j)*comb(2**(n-j)-1, i) for j in range(n+1) for i in range(1,k+1)) # _John Tyler Rascoe_, Mar 06 2025
%Y A369950 Cf. A055154, A003465 (diagonal), A102661, A381682, A381683.
%K A369950 nonn,tabf
%O A369950 1,3
%A A369950 _Manfred Boergens_, Feb 12 2024