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A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).

%I A177257 #10 May 12 2024 11:29:37
%S A177257 0,0,0,1,8,47,258,1426,8154,48715,305012,2001719,13754692,98801976,
%T A177257 740584196,5782218745,46942426080,395607218279,3455493024350,
%U A177257 31236784338746,291836182128670,2814329123555051,27980637362452980
%N A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*A000110(j).
%C A177257 Number of blocks not consisting of consecutive integers in all partitions of the set {1,2,...,n} (a singleton is considered a block of consecutive integers). Example: a(3)=1 because in 1-2-3, 1-23, 12-3, 13-2, and 123 only the block 13 does not consist of consecutive integers.
%H A177257 Alois P. Heinz, <a href="/A177257/b177257.txt">Table of n, a(n) for n = 0..500</a>
%F A177257 a(n) = Sum_{j=0..n-1} (binomial(n,j) - (j+1))*Bell(j), where Bell(n) = A000110(n) are the Bell numbers.
%F A177257 a(n) = Sum_{k=0..floor(n/2)} k*A177256(n,k).
%F A177257 a(n) = A005493(n-1) - A177255(n).
%p A177257 with(combinat): a:= proc(n) add((binomial(n, j)-j-1)*bell(j), j = 0 .. n-1) end proc: seq(a(n), n = 0 .. 22);
%t A177257 Table[Sum[(Binomial[n,j]-j-1)BellB[j],{j,0,n-1}],{n,0,30}] (* _Harvey P. Dale_, Oct 15 2015 *)
%o A177257 (Magma)
%o A177257 A177257:= func< n | n eq 0 select 0 else (&+[(Binomial(n,j)-(j+1))*Bell(j): j in [0..n-1]]) >;
%o A177257 [A177257(n): n in [0..30]]; // _G. C. Greubel_, May 12 2024
%o A177257 (SageMath)
%o A177257 def A177257(n): return sum((binomial(n,j) -(j+1))*bell_number(j) for j in range(n))
%o A177257 [A177257(n) for n in range(31)] # _G. C. Greubel_, May 12 2024
%Y A177257 Cf. A000110, A005493, A177254, A177255, A177256.
%K A177257 nonn
%O A177257 0,5
%A A177257 _Emeric Deutsch_, May 07 2010