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A270529 Sum of the sizes of the (n+1)-th blocks in all set partitions of {1,2,...,2n+1}.

%I A270529 #13 May 23 2018 11:41:02
%S A270529 1,5,47,675,13276,334751,10354804,380797185,16262852622,792102157717,
%T A270529 43370872479317,2638621340623857,176656418678888190,
%U A270529 12910491906798508171,1022900642521227415940,87345042902079159197907,7997120745886569461943400,781580696472700788364550933
%N A270529 Sum of the sizes of the (n+1)-th blocks in all set partitions of {1,2,...,2n+1}.
%H A270529 Alois P. Heinz, <a href="/A270529/b270529.txt">Table of n, a(n) for n = 0..200</a>
%H A270529 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A270529 a(n) = A270236(2n+1,n+1).
%F A270529 a(n) ~ 2^(2*n+1/2) * n^(n-1/2) / (sqrt(Pi*(1-c)) * exp(n) * c^(n+1) * (2-c)^n), where c = -A226775 = -LambertW(-2*exp(-2)) = 0.4063757399599599... . - _Vaclav Kotesovec_, Mar 19 2016
%e A270529 a(1) = 5 = 0+1+1+2+1 = sum of the sizes of the second blocks in all A000110(3) = 5 set partitions of 3: 123, 12|3, 13|2, 1|23, 1|2|3.
%p A270529 b:= proc(n, m, k) option remember; `if`(n=0, [1, 0], add((p->p+
%p A270529       `if`(j=k, [0, p[1]], 0))(b(n-1, max(m, j), k)), j=1..m+1))
%p A270529     end:
%p A270529 a:= n-> b(2*n+1, 0, n+1)[2]:
%p A270529 seq(a(n), n=0..20);
%t A270529 b[n_, m_, k_] := b[n, m, k] = If[n == 0, {1, 0}, Sum[# + If[j == k, {0, #[[1]]}, 0]&[b[n - 1, Max[m, j], k]], {j, 1, m + 1}]];
%t A270529 a[n_] := b[2*n + 1, 0, n + 1][[2]];
%t A270529 Table[a[n], {n, 0, 20}] (* _Jean-François Alcover_, May 23 2018, translated from Maple *)
%Y A270529 Cf. A000110, A270236, A285410.
%K A270529 nonn
%O A270529 0,2
%A A270529 _Alois P. Heinz_, Mar 18 2016