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

0,3

Links

Alois P. Heinz, Table of n, a(n) for n = 0..444

Formula

E.g.f.: exp(x/(1-x))*Sum_{k>0} (1-exp(x^k/(x-1))).

Example

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.

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

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 *)

Prog

(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 */

Crossrefs

Cf. A028417, A028418, A046746, A006128, A097145, A097147, A097148.

Sequence in context: A153232 A287899 A110379 * A143020 A367238 A349331

Adjacent sequences: A097143 A097144 A097145 * A097147 A097148 A097149

Keyword

easy,nonn

Author

Vladeta Jovovic, Jul 27 2004

Extensions

a(0)=0 prepended by Alois P. Heinz, May 10 2016

Status

approved