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A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.

%I A097146 #13 Nov 05 2020 06:41:05
%S A097146 0,1,5,31,217,1781,16501,172915,1998641,25468777,352751941,5292123431,
%T A097146 85297925065,1472161501981,27039872306357,527253067633531,
%U A097146 10865963240550241,236088078855319505,5390956470528548101,129102989125943058607,3234053809095307670201,84596120521251178630981,2305894874979300173268085
%N A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.
%H A097146 Alois P. Heinz, <a href="/A097146/b097146.txt">Table of n, a(n) for n = 0..444</a>
%F A097146 E.g.f.: exp(x/(1-x))*Sum_{k>0} (1-exp(x^k/(x-1))).
%e A097146 For n=4 we have 73 sets of lists (cf. A000262): (1234) (24 ways), (123)(4) (6*4 ways), (12)(34) (3*4 ways), (12)(3)(4) (6*2 ways), (1)(2)(3)(4) (1 way); so a(4)= 24*4+24*3+12*2+12*2+1*1 = 217.
%p A097146 b:= proc(n, m) option remember; `if`(n=0, m, add(j!*
%p A097146       b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n))
%p A097146     end:
%p A097146 a:= n-> b(n, 0):
%p A097146 seq(a(n), n=0..25);  # _Alois P. Heinz_, May 10 2016
%t A097146 b[n_, m_] := b[n, m] = If[n == 0, m, Sum[j! b[n-j, Max[m, j]] Binomial[n-1, j-1], {j, 1, n}]];
%t A097146 a[n_] := b[n, 0];
%t A097146 a /@ Range[0, 25] (* _Jean-François Alcover_, Nov 05 2020, after _Alois P. Heinz_ *)
%o A097146 (PARI)
%o A097146 N=50; x='x+O('x^N);
%o A097146 egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) );
%o A097146 Vec( serlaplace(egf) ) /* show terms */
%Y A097146 Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.
%K A097146 easy,nonn
%O A097146 0,3
%A A097146 _Vladeta Jovovic_, Jul 27 2004
%E A097146 a(0)=0 prepended by _Alois P. Heinz_, May 10 2016