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A270703 Total sum of the sizes of all blocks with maximal element n in all set partitions of {1,2,...,2n-1}.

%I A270703 #16 Feb 15 2017 11:27:10
%S A270703 1,4,41,670,15717,492112,19610565,961547874,56562256041,3914022281500,
%T A270703 313638627550657,28730918805512678,2976543225606178893,
%U A270703 345587228510915829224,44615408909143456529309,6361213086726610526079402,995709801367376369056571089
%N A270703 Total sum of the sizes of all blocks with maximal element n in all set partitions of {1,2,...,2n-1}.
%C A270703 Also total sum of the sizes of all blocks with minimal element n in all set partitions of {1,2,...,2n-1}.
%H A270703 Alois P. Heinz, <a href="/A270703/b270703.txt">Table of n, a(n) for n = 1..200</a>
%H A270703 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%F A270703 a(n) = A270701(2n-1,n) = A270702(2n-1,n).
%e A270703 a(2) = 4 = 0+2+1+0+1 = sum of the sizes of all blocks with maximal element 2 in all set partitions of {1,2,3}: 123, 12|3, 13|2, 1|23, 1|2|3.
%p A270703 b:= proc(n, m, t) option remember; `if`(n=0, [1, 0], add(
%p A270703      `if`(t=1 and j<>m+1, 0, (p->p+`if`(j=-t or t=1 and j=m+1,
%p A270703       [0, p[1]], 0))(b(n-1, max(m, j), `if`(t=1 and j=m+1, -j,
%p A270703      `if`(t<0, t, `if`(t>0, t-1, 0)))))), j=1..m+1))
%p A270703     end:
%p A270703 a:= n-> b(2*n-1, 0, n)[2]:
%p A270703 seq(a(n), n=1..20);
%t A270703 b[n_, m_, t_] := b[n, m, t] = If[n==0, {1, 0}, Sum[If[t==1 && j != m+1, 0, Function[p, p+If[j == -t || t == 1 && j == m+1, {0, p[[1]]}, 0]][b[n-1, Max[m, j], If[t == 1 && j == m+1, -j, If[t<0, t, If[t>0, t-1, 0]]]]]], {j, 1, m+1}]]; a[n_] := b[2*n-1, 0, n][[2]]; Table[a[n], {n, 1, 20}] (* _Jean-François Alcover_, Feb 15 2017, translated from Maple *)
%Y A270703 Cf. A000110, A270701, A270702.
%K A270703 nonn
%O A270703 1,2
%A A270703 _Alois P. Heinz_, Mar 21 2016