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A229245 Number of set partitions of {1,...,n} with largest set of size 3.

%I A229245 #10 Dec 10 2020 17:34:00
%S A229245 1,4,20,90,420,2016,10024,51640,276980,1540440,8899176,53313624,
%T A229245 330835960,2124646720,14102514560,96622736256,682608577104,
%U A229245 4966188238080,37166169295360,285813960789280,2256147419689856,18263257380872064,151466260791609600
%N A229245 Number of set partitions of {1,...,n} with largest set of size 3.
%H A229245 Alois P. Heinz, <a href="/A229245/b229245.txt">Table of n, a(n) for n = 3..500</a>
%F A229245 E.g.f.: exp(Sum_{j=1..3} x^j/j!) - exp(Sum_{j=1..2} x^j/j!).
%p A229245 G:= proc(n, k) option remember; local j; if k>n then G(n, n)
%p A229245       elif n=0 then 1 elif k<1 then 0 else G(n-k, k);
%p A229245       for j from k-1 to 1 by -1 do %*(n-j)/j +G(n-j, k) od; % fi
%p A229245     end:
%p A229245 a:= n-> G(n,3)-G(n,2):
%p A229245 seq(a(n), n=3..30);
%t A229245 b[n_, i_] := b[n, i] = If[n==0, 1, If[i<1, 0, Sum[b[n - i j, i - 1] n!/ i!^j/(n - i j)!/j!, {j, 0, n/i}]]];
%t A229245 a[n_] := b[n, 3] - b[n, 2];
%t A229245 a /@ Range[3, 30] (* _Jean-François Alcover_, Dec 10 2020, after _Alois P. Heinz_ in A080510 *)
%Y A229245 Column k=3 of A080510.
%K A229245 nonn
%O A229245 3,2
%A A229245 _Alois P. Heinz_, Sep 17 2013