A349415 - OEIS

%I #27 Jan 04 2022 23:49:17

%S 10,60,245,840,2604,7560,20955,56100,146146,372372,931385,2293200,

%T 5569880,13368528,31751223,74709900,174324430,403700220,928512277,

%U 2122315800,4823447300,10905187800,24536675475,54962156340,122607890874,272461983780,603308682865,1331439856800

%N Number of ways an n-set can be written as the union of 2 sets each with 4 or more elements and whose intersection contains exactly 3 elements.

%C Starting at n=7, the terms in the sequence alternate one odd and 3 even.

%H <a href="/index/Rec#order_08">Index entries for linear recurrences with constant coefficients</a>, signature (12,-62,180,-321,360,-248,96,-16).

%F a(n) = Sum_{j=4..n/2+1} binomial(n,j)*binomial(j,3), n even.

%F a(n) = (Sum_{j=4..ceiling(n/2)} binomial(n,j)*binomial(j,3)) + (1/2)*binomial(n,ceiling(n/2)+1)*binomial(ceiling(n/2)+1,3), n odd.

%F From _Alois P. Heinz_, Nov 16 2021: (Start)

%F a(n) = binomial(n,3) * Stirling2(n-3,2).

%F G.f.: x^5*(8*x^6 - 48*x^5 + 124*x^4 - 180*x^3 + 145*x^2 - 60*x + 10)/((2*x-1)^4*(x-1)^4). (End)

%F E.g.f.: (1/12)*x^3*(exp(x)-1)^2.

%F a(n) = 12*a(n-1) - 62*a(n-2) + 180*a(n-3) - 321*a(n-4) + 360*a(n-5) - 248*a(n-6) + 96*a(n-7) - 16*a(n-8). - _Wesley Ivan Hurt_, Dec 03 2021

%e a(5)=10 since [5] can be written as the union of the following sets: {1,2,3,4} U {1,2,3,5}, {1,2,3,4} U {1,2,4,5}, {1,2,3,4} U {1,3,4,5}, {1,2,3,4} U {2,3,4,5}, {1,2,3,5} U {1,2,4,5}, {1,2,3,5} U {1,3,4,5},{1,2,3,5} U {2,3,4,5}, {1,2,4,5} U {1,3,4,5}, {1,2,4,5} U {2,3,4,5}, {1,3,4,5} U {2,3,4,5}.

%p a:= n-> binomial(n,3)*Stirling2(n-3,2):

%p seq(a(n), n=5..32);  # _Alois P. Heinz_, Nov 16 2021

%t nterms=50;Table[Binomial[n,3]*StirlingS2[n-3,2],{n,5,nterms+4}] (* _Paolo Xausa_, Nov 20 2021 *)

%Y Cf. A052769, A260006, A348257.

%K nonn,easy

%O 5,1

%A _Enrique Navarrete_, Nov 16 2021