This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A163353 #20 Apr 07 2025 09:53:31
%S A163353 1,1,0,1,1,0,1,4,4,1,0,1,13,44,67,56,28,8,1,0,1,40,360,1546,4144,7896,
%T A163353 11408,12866,11440,8008,4368,1820,560,120,16,1,0,1,121,2680,27550,
%U A163353 180096,866432,3308736,10453960,27991600,64472200,129002640,225783740,347370800,471435000,565722640,601080385,565722720,471435600,347373600
%N A163353 G.f.: A(x,y) = Sum_{n>=0,m>=0} (2^m-1)^n*x^n * log(1+y)^m/m!.
%C A163353 From _Manfred Boergens_, Apr 07 2025: (Start)
%C A163353 T(n,k) is the number of collections of k [n]-subsets with union=[n]; with [0] = {}.
%C A163353 For n > 0: If more than half of the subsets are drawn their union covers [n] (see Formula). - The proof is based on 2^(n-1) being the number of subsets of [n] with one fixed element of [n] missing.
%C A163353 For collections of nonempty subsets see A055154.
%C A163353 For disjoint collections of subsets see A256894.
%C A163353 For disjoint collections of nonempty subsets see A008277. (End)
%H A163353 G. C. Greubel, <a href="/A163353/b163353.txt">Table of n, a(n) for the first 11 rows, flattened</a>
%F A163353 T(n,k) = Sum_{j=0..n} (-1)^(n-j)*C(n,j)*C(2^j,k), k=0..2^n.
%F A163353 Row sums form A000371 (nondegenerate Boolean functions of n variables).
%F A163353 Main diagonal equals A134174 and is defined by the g.f.:
%F A163353 Sum_{n>=0} log(1 + (2^n-1)*x)^n/n!.
%F A163353 From _Manfred Boergens_, Apr 11 2024: (Start)
%F A163353 T(n,k) = A055154(n,k) + A055154(n,k-1) for n > 0, k > 0; A055154(n,j) are not defined for j = 0 and j = 2^n and are set = 0.
%F A163353 T(n,k) = C(2^n,k) for k > 2^(n-1).
%F A163353 T(n,k) < C(2^n,k) for k <= 2^(n-1), n > 0.
%F A163353 (Note: C(2^n,k) is the number of all k-subsets of P([n]).) (End)
%e A163353 Triangle begins:
%e A163353 1,1;
%e A163353 0,1,1;
%e A163353 0,1,4,4,1;
%e A163353 0,1,13,44,67,56,28,8,1;
%e A163353 0,1,40,360,1546,4144,7896,11408,12866,11440,8008,4368,1820,560,120,16,1;
%e A163353 ...
%t A163353 Table[Sum[(-1)^(n - j)*Binomial[n, j]*Binomial[2^j, k], {j, 0,
%t A163353 n}], {n, 0, 5}, {k, 0, 2^n}]//Flatten (* _G. C. Greubel_, Dec 19 2016 *)
%o A163353 (PARI) T(n,k)=sum(j=0,n,(-1)^(n-j)*binomial(n,j)*binomial(2^j,k))
%Y A163353 Cf. A000371 (row sums), A381683 (partial row sums), A134174 (main diagonal).
%Y A163353 Cf. A008277, A055154, A256894.
%K A163353 nonn,tabf
%O A163353 0,8
%A A163353 _Paul D. Hanna_, Jul 25 2009