A087648 - OEIS

%I #23 Sep 08 2022 08:45:11

%S 1,3,9,31,120,514,2407,12205,66491,386699,2388096,15589732,107165081,

%T 773106715,5836100685,45981026703,377230766908,3215977070706,

%U 28437411817135,260380616093533,2464930698184351,24091925888687459,242802079705721156,2520198597834860148

%N a(n) = (1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)).

%C Sum of last number in all set partitions of n+1. E.g. The set partitions of 3 are {1,1,1}, {1,1,2}, {1,2,1}, {1,2,2} and {1,2,3}, so a(2) = 1+2+1+2+3 = 9. - _Franklin T. Adams-Watters_, Jun 07 2006

%C Number of partitions of the (n+2)-multiset {0,0,1,2,...,n} into distinct multisets.  Also number of factorizations of 2 * Product_{i=1..n+1} prime(i) into distinct factors. - _Alois P. Heinz_, Jul 30 2021

%H Vincenzo Librandi, <a href="/A087648/b087648.txt">Table of n, a(n) for n = 0..200</a>

%t f[0]=1; f[n_] := Sum[ StirlingS2[n, k]*Binomial[k+2, k ], {k, 1, n}]; Table[ f[n], {n, 0, 20}] (* _Zerinvary Lajos_, Mar 31 2007 *)

%t (#[[3]]+#[[2]]-#[[1]])/2&/@Partition[BellB[Range[0,30]],3,1] (* _Harvey P. Dale_, Jul 20 2021 *)

%o (Magma) [(1/2)*(Bell(n+2)+Bell(n+1)-Bell(n)) : n in [0..30]]; // _Vincenzo Librandi_, Nov 13 2011

%Y Cf. A000110, A035098, A059606.

%Y Main diagonal of A120057, row sums of A120095.

%Y Column 1 of array in A322770.

%Y Row n=2 of A346520.

%K nonn

%O 0,2

%A _Vladeta Jovovic_, Sep 23 2003