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
Examples
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}.
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).
Programs
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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 *)
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
Comments