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A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.

%I A124427 #28 Jul 28 2021 05:45:46
%S A124427 0,1,3,9,30,112,463,2095,10279,54267,306298,1838320,11677867,78207601,
%T A124427 550277003,4055549053,31224520322,250547144156,2090779592827,
%U A124427 18110124715919,162546260131455,1509352980864191,14478981877739094,143299752100925452,1461455003961745247
%N A124427 Sum of the sizes of the blocks containing the element 1 in all set partitions of {1,2,...,n}.
%H A124427 Alois P. Heinz, <a href="/A124427/b124427.txt">Table of n, a(n) for n = 0..575</a>
%F A124427 a(n) = Sum(k*binomial(n-1,k-1)*B(n-k), k=1..n) = Sum(k*A056857(n,k), k=1..n), where B(q) are the Bell numbers (A000110).
%F A124427 a(n) = (n-1)*B(n-1)+B(n). - _Vladeta Jovovic_, Nov 10 2006
%F A124427 a(n) ~ Bell(n) * (LambertW(n) + 1). - _Vaclav Kotesovec_, Jul 28 2021
%e A124427 a(3)=9 because the 5 (=A000110(3)) set partitions of {1,2,3} are 123, 12|3, 13|2, 1|23 and 1|2|3 and 3+2+2+1+1=9.
%p A124427 with(combinat): seq(add(k*binomial(n-1,k-1)*bell(n-k),k=1..n),n=0..30);
%t A124427 Table[Sum[Binomial[n-1,k-1] * BellB[n-k] * k, {k,1,n}], {n,0,22}] (* _Geoffrey Critzer_, Jun 14 2013 *)
%t A124427 Flatten[{0, Table[(n-1)*BellB[n-1] + BellB[n], {n, 1, 20}]}] (* _Vaclav Kotesovec_, Mar 19 2016, after _Vladeta Jovovic_ *)
%Y A124427 Cf. A000110, A056857.
%Y A124427 Column p=1 of A270236 or of A270702.
%Y A124427 Main diagonal of A270701.
%K A124427 nonn
%O A124427 0,3
%A A124427 _Emeric Deutsch_, Nov 10 2006
%E A124427 a(0)=0 prepended by _Alois P. Heinz_, Mar 17 2016