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A363603 Expansion of e.g.f. (1/4)*(exp(x)-x-1)*(exp(x)-1)^2.

%I A363603 #33 Nov 18 2023 18:08:57
%S A363603 3,20,90,343,1197,3966,12720,39941,123651,379132,1154790,3501219,
%T A363603 10581465,31908218,96068700,288926977,868288239,2608010424,7830584850,
%U A363603 23505386015,70544469573,211692128950,635198021640,1905845723133,5718057263067
%N A363603 Expansion of e.g.f. (1/4)*(exp(x)-x-1)*(exp(x)-1)^2.
%C A363603 4*a(n) is the number of ordered set partitions of an n-set into 3 nonempty sets such that the number of elements in a particular set (say the first one) is at least two (see example).
%C A363603 4*a(n) is also the number of ternary strings using digits {0,1,2} so that all digits are used and a particular digit appears at least twice; for example, for n=5, the 80 strings with at least two 0's are 00112 (30 of this type), 00122 (30 of this type), 00012 (20 of this type).
%H A363603 Paolo Xausa, <a href="/A363603/b363603.txt">Table of n, a(n) for n = 4..1000</a>
%H A363603 <a href="/index/Rec#order_05">Index entries for linear recurrences with constant coefficients</a>, signature (9,-31,51,-40,12).
%F A363603 a(n) = (3^n - 3*(2^n - 1))/4 - (n/2)*(2^(n-2) - 1), n>=4.
%F A363603 G.f.: x^4*(3 - 7*x + 3*x^2)/((1 - 3*x)*(1 - 2*x)^2*(1 - x)^2). - _Stefano Spezia_, Jun 11 2023
%F A363603 a(n) = (Sum_{k=2..n-2} A000225(k-1)*binomial(n,k))/2. - _R. J. Cano_, Jul 27 2023
%e A363603 4*a(5)=80 since the ordered set partitions are the following: 30 of type {1,2}{3,4},{5}; 30 of type {1,2},{3},{4,5}; 20 of type {1,2,3},{4},{5}.
%t A363603 A363603[n_]:=(3^n-3(2^n-1))/4-(n/2)(2^(n-2)-1);Array[A363603,40,4] (* or *)
%t A363603 LinearRecurrence[{9,-31,51,-40,12},{3,20,90,343,1197},40] (* _Paolo Xausa_, Nov 18 2023 *)
%Y A363603 Cf. A000225, A052749, A001117, A363591.
%K A363603 nonn,easy
%O A363603 4,1
%A A363603 _Enrique Navarrete_, Jun 11 2023