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a(n) = 2*A040027(n-1) + Bell(n), where Bell = A000110.
4

%I #31 Sep 17 2024 12:50:14

%S 1,3,4,11,33,114,445,1923,9078,46369,254297,1487896,9239135,60615819,

%T 418583924,3032405831,22979752405,181697363626,1495586215841,

%U 12789423056183,113415288869750,1041244540823413,9881851825756365,96811870321650792,977851660102425867

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

%H Paolo Xausa, <a href="/A038559/b038559.txt">Table of n, a(n) for n = 0..500</a>

%H 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 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 := proc(n)

%p 2*A040027(n-1)+combinat[bell](n) ;

%p end proc: # _R. J. Mathar_, Dec 20 2013

%p alias(PS = ListTools:-PartialSums):

%p A038559List := proc(len) local a, k, P, T; a := 3; P := [1]; T := [1];

%p for k from 1 to len-1 do

%p T := [op(T), a]; P := PS([a, op(P)]); a := P[-1] od;

%p T end: A038559List(25); # _Peter Luschny_, Mar 28 2022

%t a[0] = 1; a[1] = 3; a[n_] := a[n] = Sum[Binomial[n - 1, k - 1]*a[n - k], {k, n}];

%t Array[a, 30, 0] (* _Paolo Xausa_, Sep 17 2024 *)

%t 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 Cf. A000110, A040027.

%K easy,nonn

%O 0,2

%A _Henry Gould_