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a(n) = n*B(n-1) + n*(n-1)*B(n-2), where the B(i) are Bell numbers (A000110).
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%I #17 Sep 08 2022 08:46:05

%S 0,1,4,12,44,175,762,3605,18384,100404,584070,3601895,23451540,

%T 160633681,1153896772,8668821600,67943174000,554327140739,

%U 4698491153454,41299244789989,375844030441560

%N a(n) = n*B(n-1) + n*(n-1)*B(n-2), where the B(i) are Bell numbers (A000110).

%H Vincenzo Librandi, <a href="/A226855/b226855.txt">Table of n, a(n) for n = 0..200</a>

%H B. Chern, P. Diaconis, D. M. Kane, R. C. Rhoades, <a href="http://math.stanford.edu/~rhoades/FILES/setpartitions.pdf">Closed expressions for averages of set partition statistics</a>, 2013.

%t Table[n BellB[n-1] + n (n-1) BellB[n-2], {n, 0, 30}] (* _Vincenzo Librandi_, Jul 16 2013 *)

%o (PARI) B(n) = if (n<=1, return (1), return (sum(i=0, n-1, binomial(n-1, i)*B(n-1-i))))

%o a(n) = n*B(n-1) + n*(n-1)*B(n-2)

%o (Magma) [0,1] cat [n*Bell(n-1)+n*(n-1)*Bell(n-2): n in [2..25]]; // _Vincenzo Librandi_, Jul 16 2013

%Y Cf. A052889 (see Prop 3.1 (ii) in Chern et al. link).

%K nonn

%O 0,3

%A _Michel Marcus_, Jun 19 2013