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%I #35 Dec 07 2021 16:40:17
%S 6,30,105,315,868,2268,5715,14025,33726,79794,186277,429975,982920,
%T 2228088,5013351,11206485,24903490,55050030,121110297,265289475,
%U 578813676,1258290900,2726297275,5888802465,12683574918,27246198378,58384711245,124822486575,266287971856,566935682544
%N Number of ways we can write [n] as the union of 2 sets of sizes i, j which intersect in exactly 2 elements (2 < i,j < n; i = j allowed).
%C The terms in the sequence alternate 2 even and 2 odd.
%H <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F a(n) = (Sum_{j=3..n/2} binomial(n,j)*binomial(j,2)) + (1/2)*binomial(n,n/2+1) * binomial(n/2+1,2), if n is even.
%F a(n) = Sum_{j=3..ceiling(n/2)} binomial(n,j)*binomial(j,2), if n is odd.
%F G.f.: x^4*(6 - 24*x + 33*x^2 - 18*x^3 + 4*x^4)/((1 - x)^3*(1 - 2*x)^3). - _Stefano Spezia_, Oct 09 2021
%F a(n) = A000554(n)/2. - _Enrique Navarrete_, Nov 16 2021
%F a(n) = binomial(n,2) * Stirling2(n-2,2). - _Alois P. Heinz_, Nov 16 2021
%e a(4) = 6 since we can write [4] as the following unions: {1,2,3} U {1,2,4}, {1,2,3} U {1,3,4}, {1,2,3} U {2,3,4}, {1,2,4} U {1,3,4}, {1,2,4} U {2,3,4}, {1,3,4} U {2,3,4}.
%t nterms=50;Table[Binomial[n,2]*StirlingS2[n-2,2],{n,4,nterms+3}] (* _Paolo Xausa_, Nov 20 2021 *)
%Y Cf. A000554, A260006.
%K nonn,easy
%O 4,1
%A _Enrique Navarrete_, Oct 08 2021