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A363603
Expansion of e.g.f. (1/4)*(exp(x)-x-1)*(exp(x)-1)^2.
2
3, 20, 90, 343, 1197, 3966, 12720, 39941, 123651, 379132, 1154790, 3501219, 10581465, 31908218, 96068700, 288926977, 868288239, 2608010424, 7830584850, 23505386015, 70544469573, 211692128950, 635198021640, 1905845723133, 5718057263067
OFFSET
4,1
COMMENTS
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).
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).
FORMULA
a(n) = (3^n - 3*(2^n - 1))/4 - (n/2)*(2^(n-2) - 1), n>=4.
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
a(n) = (Sum_{k=2..n-2} A000225(k-1)*binomial(n,k))/2. - R. J. Cano, Jul 27 2023
EXAMPLE
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}.
MATHEMATICA
A363603[n_]:=(3^n-3(2^n-1))/4-(n/2)(2^(n-2)-1); Array[A363603, 40, 4] (* or *)
LinearRecurrence[{9, -31, 51, -40, 12}, {3, 20, 90, 343, 1197}, 40] (* Paolo Xausa, Nov 18 2023 *)
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Enrique Navarrete, Jun 11 2023
STATUS
approved