A097146 Total sum of maximum list sizes in all sets of lists of n-set, cf. A000262.
0, 1, 5, 31, 217, 1781, 16501, 172915, 1998641, 25468777, 352751941, 5292123431, 85297925065, 1472161501981, 27039872306357, 527253067633531, 10865963240550241, 236088078855319505, 5390956470528548101, 129102989125943058607, 3234053809095307670201, 84596120521251178630981, 2305894874979300173268085
Offset: 0
Examples
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.
Links
- Alois P. Heinz, Table of n, a(n) for n = 0..444
Programs
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Maple
b:= proc(n, m) option remember; `if`(n=0, m, add(j!* b(n-j, max(m, j))*binomial(n-1, j-1), j=1..n)) end: a:= n-> b(n, 0): seq(a(n), n=0..25); # Alois P. Heinz, May 10 2016
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Mathematica
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}]]; a[n_] := b[n, 0]; a /@ Range[0, 25] (* Jean-François Alcover, Nov 05 2020, after Alois P. Heinz *)
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PARI
N=50; x='x+O('x^N); egf=exp(x/(1-x))*sum(k=1,N, (1-exp(x^k/(x-1))) ); Vec( serlaplace(egf) ) /* show terms */
Formula
E.g.f.: exp(x/(1-x))*Sum_{k>0} (1-exp(x^k/(x-1))).
Extensions
a(0)=0 prepended by Alois P. Heinz, May 10 2016