A349415 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.
10, 60, 245, 840, 2604, 7560, 20955, 56100, 146146, 372372, 931385, 2293200, 5569880, 13368528, 31751223, 74709900, 174324430, 403700220, 928512277, 2122315800, 4823447300, 10905187800, 24536675475, 54962156340, 122607890874, 272461983780, 603308682865, 1331439856800
Offset: 5
Examples
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}.
Links
- Index entries for linear recurrences with constant coefficients, signature (12,-62,180,-321,360,-248,96,-16).
Programs
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Maple
a:= n-> binomial(n,3)*Stirling2(n-3,2): seq(a(n), n=5..32); # Alois P. Heinz, Nov 16 2021
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Mathematica
nterms=50;Table[Binomial[n,3]*StirlingS2[n-3,2],{n,5,nterms+4}] (* Paolo Xausa, Nov 20 2021 *)
Formula
a(n) = Sum_{j=4..n/2+1} binomial(n,j)*binomial(j,3), n even.
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.
From Alois P. Heinz, Nov 16 2021: (Start)
a(n) = binomial(n,3) * Stirling2(n-3,2).
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)
E.g.f.: (1/12)*x^3*(exp(x)-1)^2.
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
Comments