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

Links

Paolo Xausa, Table of n, a(n) for n = 4..1000

Index entries for linear recurrences with constant coefficients, signature (9,-31,51,-40,12).

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

Cf. A000225, A052749, A001117, A363591.

Sequence in context: A171673 A185065 A227811 * A092786 A015529 A278319

Adjacent sequences: A363600 A363601 A363602 * A363604 A363605 A363606

Keyword

nonn,easy

Author

Enrique Navarrete, Jun 11 2023

Status

approved