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A114917 Number of partitions of {1,..,n} in which no part of a given size occurs just once.

%I A114917 #11 May 26 2023 08:49:21
%S A114917 1,0,1,1,4,1,71,106,1051,2759,19552,51041,864579,3134132,34990671,
%T A114917 211464345,1832236004,11261632321,109973219879,659853699654,
%U A114917 8379292553185,58134013363151,676374746166550,5912498819726335,71622214447120275,658455096592878092
%N A114917 Number of partitions of {1,..,n} in which no part of a given size occurs just once.
%H A114917 Alois P. Heinz, <a href="/A114917/b114917.txt">Table of n, a(n) for n = 0..576</a>
%p A114917 b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
%p A114917       add(`if`(j=1, 0, b(n-i*j, i-1)*combinat[multinomial]
%p A114917               (n, n-i*j, i$j)/j!), j=0..min(n/i))))
%p A114917     end:
%p A114917 a:= n-> b(n$2):
%p A114917 seq(a(n), n=0..26);  # _Alois P. Heinz_, May 14 2023
%t A114917 multinomial[n_, k_List] := n!/Times @@ (k!);
%t A114917 b[n_, i_] := b[n, i] = If[n == 0, 1, If[i < 1, 0, Sum[If[j == 1, 0, b[n - i*j, i - 1]*multinomial[n, Prepend[Table[i, {j}], n - i*j]]/j!], {j, 0, Min[n/i]}]]];
%t A114917 a[n_] := b[n, n];
%t A114917 Table[a[n], {n, 0, 26}] (* _Jean-François Alcover_, May 26 2023, after _Alois P. Heinz_ *)
%Y A114917 Cf. A000110, A007690, A360182.
%K A114917 nonn
%O A114917 0,5
%A A114917 _Christian G. Bower_, Jan 06 2006