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A038559 a(n) = 2*A040027(n-1) + Bell(n), where Bell = A000110.

%I A038559 #31 Sep 17 2024 12:50:14
%S A038559 1,3,4,11,33,114,445,1923,9078,46369,254297,1487896,9239135,60615819,
%T A038559 418583924,3032405831,22979752405,181697363626,1495586215841,
%U A038559 12789423056183,113415288869750,1041244540823413,9881851825756365,96811870321650792,977851660102425867
%N A038559 a(n) = 2*A040027(n-1) + Bell(n), where Bell = A000110.
%H A038559 Paolo Xausa, <a href="/A038559/b038559.txt">Table of n, a(n) for n = 0..500</a>
%H A038559 H. W. Gould and J. Quaintance, <a href="https://doi.org/10.2298/AADM0702371G">A linear binomial recurrence and the Bell numbers and polynomials</a>, Applic. Anal. Discr. Math 1 (2007) 371-385.
%F A038559 a(0) = 1, a(1) = 3; a(n) = Sum_{k=1..n} binomial(n-1,k-1) * a(n-k). - _Ilya Gutkovskiy_, Jul 10 2020
%p A038559 A038559 := proc(n)
%p A038559     2*A040027(n-1)+combinat[bell](n) ;
%p A038559 end proc: # _R. J. Mathar_, Dec 20 2013
%p A038559 alias(PS = ListTools:-PartialSums):
%p A038559 A038559List := proc(len) local a, k, P, T; a := 3; P := [1]; T := [1];
%p A038559 for k from 1 to len-1 do
%p A038559    T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;
%p A038559 T end: A038559List(25); # _Peter Luschny_, Mar 28 2022
%t A038559 a[0] = 1; a[1] = 3; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1]*a[n - k], {k, n}];
%t A038559 Array[a, 30, 0] (* _Paolo Xausa_, Sep 17 2024 *)
%t A038559 Module[{a = 3, p = {1}}, Join[{1, a}, Table[a = Last[p = Accumulate[Prepend[p, a]]], 28]]] (* _Paolo Xausa_, Sep 17 2024, after _Peter Luschny_ *)
%Y A038559 Cf. A000110, A040027.
%K A038559 easy,nonn
%O A038559 0,2
%A A038559 _Henry Gould_