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Number of ternary strings of length n that contain at least two 0's and at most two 1's.
2

%I #18 Feb 15 2021 23:04:35

%S 0,0,1,7,33,121,378,1065,2803,7045,17148,40789,95373,220065,502414,

%T 1136977,2553831,5699149,12645504,27914877,61337665,134213065,

%U 292547346,635430937,1375724763,2969559381,6392110468,13723752805,29393671413,62813884465,133949278998,285078439329,605590372303

%N Number of ternary strings of length n that contain at least two 0's and at most two 1's.

%H <a href="/index/Rec#order_07">Index entries for linear recurrences with constant coefficients</a>, signature (10,-42,96,-129,102,-44,8).

%F a(n) = 2^n + n*2^(n-1) + binomial(n,2)*2^(n-2) - 3*binomial(n,2) - 3*binomial(n,3) - 2*n - 1.

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

%F G.f.: x^2*(1-3*x+5*x^2-11*x^3+11*x^4)/((1-x)^4*(1-2*x)^3). - _Stefano Spezia_, Jan 31 2021

%e a(4) = 33 since the strings are composed of 0000, the 4 permutations of 0001, the 4 permutations of 0002, the 6 permutations of 0011, the 6 permutations of 0022, and the 12 permutations of 0012. Thus, the total number of strings is 1 + 4 + 4 + 6 + 6 + 12 = 33.

%t CoefficientList[Series[Exp[x](Exp[x]-1-x)(2+2x+x^2)/2,{x,0,32}],x]Table[i!,{i,0,32}] (* _Stefano Spezia_, Jan 31 2021 *)

%Y Cf. A338229, A338230.

%K nonn,easy

%O 0,4

%A _Enrique Navarrete_, Jan 30 2021