A051303 Number of 3-element proper antichains of an n-element set.
0, 0, 0, 1, 30, 605, 9030, 110901, 1200150, 11932285, 111885510, 1006471301, 8786447670, 75039565965, 630534185190, 5234341175701, 43059373189590, 351805681631645, 2859550165976070, 23152657123816101, 186907026783617910, 1505512392025329325
Offset: 0
Links
- G. C. Greubel, Table of n, a(n) for n = 0..1000
- Index entries for linear recurrences with constant coefficients, signature (29,-343,2135,-7504,14756,-14832,5760).
Programs
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Magma
[(8^n - 9*6^n + 15*5^n - 4*4^n - 9*3^n + 8*2^n - 2) / Factorial(3) : n in [0..25]]; // G. C. Greubel, Oct 06 2017
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Maple
A051303:=n->(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!: seq(A051303(n), n=0..30); # Wesley Ivan Hurt, Oct 06 2017
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Mathematica
Table[(8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!, {n,0,25}] (* G. C. Greubel, Oct 06 2017 *) -
PARI
for(n=0,25, print1((8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2 )/3!, ", ")) \\ G. C. Greubel, Oct 06 2017
Formula
a(n) = (8^n -9*6^n +15*5^n -4*4^n -9*3^n +8*2^n -2)/3!.
G.f.: x^3*(360*x^3-78*x^2-x-1) / ((x-1)*(2*x-1)*(3*x-1)*(4*x-1)*(5*x-1)*(6*x-1)*(8*x-1)). - Colin Barker, Nov 27 2012
a(n) = 29*a(n-1) - 343*a(n-2) + 2135*a(n-3) - 7504*a(n-4) + 14756*a(n-5) - 14832*a(n-6) + 5760*a(n-7) for n > 6. - Wesley Ivan Hurt, Oct 06 2017
E.g.f.: exp(x)*(exp(x) - 1)^3*(2 - 2*exp(x) - 3*exp(2*x) + 3*exp(3*x) + exp(4*x))/6. - Stefano Spezia, Sep 28 2024