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Number of 3-block covers of a labeled n-set.
1

%I #29 Mar 18 2024 16:46:56

%S 1,32,321,2560,18881,135072,954241,6705920,47020161,329377312,

%T 2306349761,16146574080,113032395841,791245902752,5538778714881,

%U 38771623191040,271401878897921,1899814701967392,13298707562817601,93090966886860800,651636810049438401

%N Number of 3-block covers of a labeled n-set.

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (11,-31,21).

%F a(n) = (1/3!)*(11-6*3^n+7^n).

%F a(n) = 11*a(n-1)-31*a(n-2)+21*a(n-3). G.f.: -x^2*(21*x+1) / ((x-1)*(3*x-1)*(7*x-1)). - _Colin Barker_, Jul 12 2013

%F a(n) = sum(i=0..n, (-1)^i * C(n,i) * C(2^(n-i)-1,3) ). - _Geoffrey Critzer_, Aug 24 2014

%p seq((11-6*3^n+7^n)/6, n=2..50); # _Robert Israel_, Aug 25 2014

%t nn = 19; Table[Sum[(-1)^i Binomial[n, i] Binomial[2^(n - i) - 1, 3], {i, 0, n}], {n, 2, nn}] (* _Geoffrey Critzer_, Aug 24 2014 *)

%t Table[(11 - 6*3^n + 7^n)/6, {n, 2, 20}] (* _Wesley Ivan Hurt_, Aug 26 2014 *)

%o (Magma) [(11-6*3^n+7^n)/6 : n in [2..30]]; // _Wesley Ivan Hurt_, Aug 26 2014

%Y Column of A055154.

%K easy,nonn

%O 2,2

%A _Vladeta Jovovic_, May 31 2004

%E More terms from _Colin Barker_, Jul 12 2013