cp's OEIS Frontend

A348257 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).

%I A348257 #35 Dec 07 2021 16:40:17
%S A348257 6,30,105,315,868,2268,5715,14025,33726,79794,186277,429975,982920,
%T A348257 2228088,5013351,11206485,24903490,55050030,121110297,265289475,
%U A348257 578813676,1258290900,2726297275,5888802465,12683574918,27246198378,58384711245,124822486575,266287971856,566935682544
%N A348257 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 A348257 The terms in the sequence alternate 2 even and 2 odd.
%H A348257 <a href="/index/Rec#order_06">Index entries for linear recurrences with constant coefficients</a>, signature (9,-33,63,-66,36,-8).
%F A348257 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 A348257 a(n) = Sum_{j=3..ceiling(n/2)} binomial(n,j)*binomial(j,2), if n is odd.
%F A348257 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 A348257 a(n) = A000554(n)/2. - _Enrique Navarrete_, Nov 16 2021
%F A348257 a(n) = binomial(n,2) * Stirling2(n-2,2). - _Alois P. Heinz_, Nov 16 2021
%e A348257 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 A348257 nterms=50;Table[Binomial[n,2]*StirlingS2[n-2,2],{n,4,nterms+3}] (* _Paolo Xausa_, Nov 20 2021 *)
%Y A348257 Cf. A000554, A260006.
%K A348257 nonn,easy
%O A348257 4,1
%A A348257 _Enrique Navarrete_, Oct 08 2021