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%I #8 Nov 01 2024 09:36:14
%S 0,0,2,12,62,290,1292,5579,23606,98490,406862,1668689,6807704,
%T 27663441,112076057,453031502,1828018406,7366128866,29650536878,
%U 119249689265,479277846962,1925216817095,7729973578307,31025341749680,124486445913728,499362094315865
%N Number of connected pairs of subsets of [n] with each being a different size.
%C Empirically, a(A075930(n)) == 1 (mod 2).
%F a(n) = Sum_{i=0..n-2} binomial(n,i) * Sum_{j=i+1..n-1} (binomial(n,j) - binomial(i,n-j)).
%e a(3) = 12 counts the pairs: {{1,2},{1}}, {{1,2},{2}}, {{1,3},{1}}, {{1,3},{3}}, {{2,3},{2}}, {{2,3},{3}}, {{1,2,3},{1,2}}, {{1,2,3},{1,3}}, {{1,2,3},{2,3}}, {{1,2,3},{1}}, {{1,2,3},{2}}, {{1,2,3},{3}}.
%o (PARI)
%o A377464(n) = {sum(i=0,n-2,binomial(n,i)*sum(j=i+1,n-1, binomial(n,j)-binomial(i,n-j)))}
%Y Cf. A001187, A075930, A323818, A326749.
%K nonn,easy
%O 0,3
%A _John Tyler Rascoe_, Oct 29 2024