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%I #20 May 12 2024 02:02:29
%S 0,1,3,9,29,104,416,1837,8853,46113,257583,1533308,9676148,64452909,
%T 451475027,3314964857,25442301577,203604718076,1695172374548,
%U 14654631691569,131309475792709,1217516798735521,11664652754184043,115319114738472472,1174967255260496776
%N a(n) = Sum_{j=1..n} j*B(j-1), where B(k) = A000110(k) are the Bell numbers.
%C Number of adjacent blocks in all partitions of the set {1,2,...,n}. An adjacent block is a block of the form (i, i+1, i+2, ...). Example: a(3)=9 because in 1-2-3, 1-23, 12-3, 13-2, and 123 we have 3, 2, 2, 1, and 1 adjacent blocks, respectively.
%H Harvey P. Dale, <a href="/A177255/b177255.txt">Table of n, a(n) for n = 0..575</a>
%F a(n) = Sum_{k=0..n} k * A177254(n,k).
%p with(combinat): a := proc (n) options operator, arrow: sum(j*bell(j-1), j = 1 .. n) end proc; seq(a(n), n = 0 .. 23);
%t With[{nn=30},Join[{0},Accumulate[BellB[Range[0,nn-1]]Range[nn]]]] (* _Harvey P. Dale_, Nov 10 2014 *)
%o (Magma)
%o [n eq 0 select 0 else (&+[j*Bell(j-1): j in [1..n]]): n in [0..30]]; // _G. C. Greubel_, May 11 2024
%o (SageMath)
%o [sum(j*bell_number(j-1) for j in range(1,1+n)) for n in range(31)] # _G. C. Greubel_, May 11 2024
%Y Cf. A000110, A177254, A177256, A177257.
%Y Partial sums of A052889.
%K nonn
%O 0,3
%A _Emeric Deutsch_, May 07 2010