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 A340135 #9 Dec 22 2024 16:24:23
%S A340135 0,0,0,0,24,0,720,0,7000,15120,126000,0,1777776,0,23543520,55855800,
%T A340135 274565720,0,5337775872,0,63026049424,117920013120,995265791520,0,
%U A340135 15265486117744,14283091977000,216344919117600,240142901941800,2854493961432480,0,55689696384165720
%N A340135 Number of pairs of independent nontrivial subsets of a finite set composed of n elements.
%C A340135 A subset X of a set S is called a trivial subset, if it is equal to the empty set or to the full set S. The subsets A and B of a finite set S are called independent, if #A/#S * #B/#S = #(A \intersect B)/#S.
%H A340135 Jochen Ziegenbalg, <a href="https://jochen-ziegenbalg.github.io/materialien/Manuskripte/independent-subsets.pdf">Independent subsets</a>
%e A340135 For n=4 and S={1,2,3,4} the a(4)=24 pairs of independent nontrivial subsets of S are
%e A340135 {{1, 2}, {1, 3}}, {{1, 2}, {1, 4}}, {{1, 2}, {2, 3}}, {{1, 2}, {2, 4}},
%e A340135 {{1, 3}, {1, 2}}, {{1, 3}, {1, 4}}, {{1, 3}, {2, 3}}, {{1, 3}, {3, 4}},
%e A340135 {{1, 4}, {1, 2}}, {{1, 4}, {1, 3}}, {{1, 4}, {2, 4}}, {{1, 4}, {3, 4}},
%e A340135 {{2, 3}, {1, 2}}, {{2, 3}, {1, 3}}, {{2, 3}, {2, 4}}, {{2, 3}, {3, 4}},
%e A340135 {{2, 4}, {1, 2}}, {{2, 4}, {1, 4}}, {{2, 4}, {2, 3}}, {{2, 4}, {3, 4}},
%e A340135 {{3, 4}, {1, 3}}, {{3, 4}, {1, 4}}, {{3, 4}, {2, 3}}, {{3, 4}, {2, 4}}
%e A340135 Tables:
%e A340135 n all independent independent
%e A340135 independent proper nontrivial
%e A340135 subsets subsets subsets
%e A340135 (see A121312) (see A158345) a(n)
%e A340135 0 1 0 0
%e A340135 1 4 1 0
%e A340135 2 12 5 0
%e A340135 3 28 13 0
%e A340135 4 84 53 24
%e A340135 5 124 61 0
%e A340135 6 972 845 720
%e A340135 7 508 253 0
%e A340135 8 8020 7509 7000
%e A340135 9 17164 16141 15120
%e A340135 10 130092 128045 126000
%e A340135 11 8188 4093 0
%e A340135 12 1794156 1785965 1777776
%e A340135 13 32764 16381 0
%e A340135 14 23609052 23576285 23543520
%e A340135 15 55986868 55921333 55855800
%e A340135 16 274827860 274696789 274565720
%e A340135 17 524284 262141 0
%e A340135 18 5338824444 5338300157 5337775872
%e A340135 19 2097148 1048573 0
%e A340135 20 63030243724 63028146573 63026049424
%e A340135 21 117928401724 117924207421 117920013120
%e A340135 22 995282568732 995274180125 995265791520
%e A340135 23 33554428 16777213 0
%e A340135 24 15265553226604 15265519672173 15265486117744
%e A340135 25 14283226194724 14283159085861 14283091977000
%e A340135 26 216345187553052 216345053335325 216344919117600
%e A340135 27 240143438812708 240143170377253 240142901941800
%e A340135 28 2854495035174300 2854494498303389 2854493961432480
%e A340135 29 2147483644 1073741821 0
%e A340135 30 55689700679133012 55689698531649365 55689696384165720
%o A340135 (Maxima) /* version 1 */
%o A340135 pairs_independent_nontrivial_subsets(n) :=
%o A340135 block([a, b, d, s : 0 ],
%o A340135 for a:1 thru n-1 do
%o A340135 for d:1 thru a do
%o A340135 ( b : n*d / a,
%o A340135 if integerp(b) and b<n
%o A340135 then (s : s + binomial(n,a)*binomial(a,d)*binomial(n-a,b-d) ) ),
%o A340135 s );
%o A340135 (Maxima) /* version 2 */
%o A340135 a(n) :=
%o A340135 sum(
%o A340135 sum(
%o A340135 (b : n*d / a,
%o A340135 if integerp(b) and b<n then
%o A340135 binomial(n,a)*binomial(a,d)*binomial(n-a,b-d) else 0), d,1,a), a,1,n-1) ;
%Y A340135 Cf. A121312 (independent subsets), A158345 (independent proper subsets).
%K A340135 nonn
%O A340135 0,5
%A A340135 _Jochen Ziegenbalg_, Dec 29 2020