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A332942 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by increasing lengths.

%I A332942 #10 Jan 06 2021 02:12:35
%S A332942 1,7,25,101,366,1555,7099,34627,184033,1059972,6425992,41266681,
%T A332942 280938451,2009636335,15025372685,117386912433,956458929950,
%U A332942 8104399834719,71244441818927,648761935841876,6110827367541999,59454153443971106,596654820386392152
%N A332942 Number of entries in the second blocks of all set partitions of [n] when blocks are ordered by increasing lengths.
%H A332942 Alois P. Heinz, <a href="/A332942/b332942.txt">Table of n, a(n) for n = 2..576</a>
%H A332942 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%p A332942 b:= proc(n, i, t) option remember; `if`(n=0, [1, 0], `if`(i>n, 0,
%p A332942       add((p-> p+`if`(t>0 and t-j<1, [0, p[1]*i], 0))(b(n-i*j, i+1,
%p A332942       max(0, t-j))/j!*combinat[multinomial](n, i$j, n-i*j)), j=0..n/i)))
%p A332942     end:
%p A332942 a:= n-> b(n, 1, 2)[2]:
%p A332942 seq(a(n), n=2..24);
%t A332942 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A332942 b[n_, i_, t_] := b[n, i, t] = If[n == 0, {1, 0}, If[i > n, 0 {0, 0}, Sum[ Function[p, p + If[t > 0 && t - j < 1, {0, p[[1]]*i}, {0, 0}]][b[n - i*j, i + 1, Max[0, t - j]]/j!*multinomial[n, Append[Table[i, {j}], n - i*j]]], {j, 0, n/i}]]];
%t A332942 a[n_] := b[n, 1, 2][[2]];
%t A332942 a /@ Range[2, 24] (* _Jean-François Alcover_, Jan 06 2021, after _Alois P. Heinz_ *)
%Y A332942 Column k=2 of A319298.
%K A332942 nonn
%O A332942 2,2
%A A332942 _Alois P. Heinz_, Mar 03 2020