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 A136556 #32 Jan 02 2024 15:18:20
%S A136556 1,1,3,35,1365,169911,67945521,89356415775,396861704798625,
%T A136556 6098989894499557055,331001552386330913728641,
%U A136556 64483955378425999076128999167,45677647585984911164223317311276545,118839819203635450208125966070067352769535,1144686912178270649701033287538093722740144666625
%N A136556 a(n) = binomial(2^n - 1, n).
%C A136556 Number of n x n binary matrices without zero rows and with distinct rows up to permutation of rows, cf. A014070.
%C A136556 Row 0 of square array A136555.
%C A136556 From _Gus Wiseman_, Dec 19 2023: (Start)
%C A136556 Also the number of n-element sets of nonempty subsets of {1..n}, or set-systems with n vertices and n edges (not necessarily covering). The covering case is A054780. For example, the a(3) = 35 set-systems are:
%C A136556 {1}{2}{3} {1}{2}{12} {1}{2}{123} {1}{12}{123} {12}{13}{123}
%C A136556 {1}{2}{13} {1}{3}{123} {1}{13}{123} {12}{23}{123}
%C A136556 {1}{2}{23} {1}{12}{13} {1}{23}{123} {13}{23}{123}
%C A136556 {1}{3}{12} {1}{12}{23} {2}{12}{123}
%C A136556 {1}{3}{13} {1}{13}{23} {2}{13}{123}
%C A136556 {1}{3}{23} {2}{3}{123} {2}{23}{123}
%C A136556 {2}{3}{12} {2}{12}{13} {3}{12}{123}
%C A136556 {2}{3}{13} {2}{12}{23} {3}{13}{123}
%C A136556 {2}{3}{23} {2}{13}{23} {3}{23}{123}
%C A136556 {3}{12}{13} {12}{13}{23}
%C A136556 {3}{12}{23}
%C A136556 {3}{13}{23}
%C A136556 Of these, only {{1},{2},{1,2}}, {{1},{3},{1,3}}, and {{2},{3},{2,3}} do not cover the vertex set.
%C A136556 (End)
%H A136556 G. C. Greubel, <a href="/A136556/b136556.txt">Table of n, a(n) for n = 0..50</a>
%F A136556 a(n) = Sum_{k=0..n} (-1)^(n-k) * binomial(2^n,k).
%F A136556 a(n) = (1/n!)*Sum_{k=0..n} Stirling1(n,k) * (2^n-1)^k.
%F A136556 G.f.: Sum_{n>=0} log(1 + 2^n*x)^n / (n! * (1 + 2^n*x)).
%F A136556 a(n) ~ 2^(n^2)/n!. - _Vaclav Kotesovec_, Jul 02 2016
%e A136556 G.f.: A(x) = 1 + x + 3*x^2 + 35*x^3 + 1365*x^4 + 169911*x^5 +...
%e A136556 A(x) = 1/(1+x) + log(1+2*x)/(1+2*x) + log(1+4*x)^2/(2!*(1+4*x)) + log(1+8*x)^3/(3!*(1+8*x)) + log(1+16*x)^4/(4!*(1+16*x)) + log(1+32*x)^5/(5!*(1+32*x)) +...
%p A136556 A136556:= n-> binomial(2^n-1,n); seq(A136556(n), n=0..20); # _G. C. Greubel_, Mar 14 2021
%t A136556 f[n_] := Binomial[2^n - 1, n]; Array[f, 12] (* _Robert G. Wilson v_ *)
%t A136556 Table[Length[Subsets[Rest[Subsets[Range[n]]],{n}]],{n,0,4}] (* _Gus Wiseman_, Dec 19 2023 *)
%o A136556 (PARI) {a(n) = binomial(2^n-1,n)}
%o A136556 for(n=0, 20, print1(a(n), ", "))
%o A136556 (PARI) /* As coefficient of x^n in the g.f.: */
%o A136556 {a(n) = polcoeff( sum(i=0,n, 1/(1 + 2^i*x +x*O(x^n)) * log(1 + 2^i*x +x*O(x^n))^i/i!), n)}
%o A136556 for(n=0, 20, print1(a(n), ", "))
%o A136556 (Sage) [binomial(2^n -1, n) for n in (0..20)] # _G. C. Greubel_, Mar 14 2021
%o A136556 (Magma) [Binomial(2^n -1, n): n in [0..20]]; // _G. C. Greubel_, Mar 14 2021
%o A136556 (Python)
%o A136556 from math import comb
%o A136556 def A136556(n): return comb((1<<n)-1,n) # _Chai Wah Wu_, Jan 02 2024
%Y A136556 Sequences of the form binomial(2^n +p*n +q, n): this sequence (0,-1), A014070 (0,0), A136505 (0,1), A136506 (0,2), A060690 (1,-1), A132683 (1,0), A132684 (1,1), A132685 (2,0), A132686 (2,1), A132687 (3,-1), A132688 (3,0), A132689 (3,1).
%Y A136556 Cf. A066384, A136555, A136557.
%Y A136556 The covering case A054780 has binomial transform A367916, ranks A367917.
%Y A136556 Connected graphs of this type are A057500, unlabeled A001429.
%Y A136556 Graphs of this type are A116508, covering A367863, unlabeled A006649.
%Y A136556 A003465 counts set-systems covering {1..n}, unlabeled A055621.
%Y A136556 A058891 counts set-systems, connected A323818, without singletons A016031.
%Y A136556 Cf. A055154, A059201, A133686, A367862, A367902, A367903.
%K A136556 nonn
%O A136556 0,3
%A A136556 _Paul D. Hanna_, Jan 07 2008; Paul Hanna and _Vladeta Jovovic_, Jan 15 2008
%E A136556 Edited by _N. J. A. Sloane_, Jan 26 2008