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A086659 T(n,k) counts the set partitions of n containing k-1 blocks of length 1.

%I A086659 #16 Jan 05 2022 18:16:14
%S A086659 1,1,3,4,4,6,11,20,10,10,41,66,60,20,15,162,287,231,140,35,21,715,
%T A086659 1296,1148,616,280,56,28,3425,6435,5832,3444,1386,504,84,36,17722,
%U A086659 34250,32175,19440,8610,2772,840,120,45,98253,194942,188375,117975,53460,18942,5082,1320,165,55
%N A086659 T(n,k) counts the set partitions of n containing k-1 blocks of length 1.
%H A086659 Alois P. Heinz, <a href="/A086659/b086659.txt">Rows n = 2..142, flattened</a>
%F A086659 E.g.f.: exp(x*y)*(exp(exp(x)-1-x)-1). - _Vladeta Jovovic_, Jul 28 2003
%e A086659 The 15 set partitions of {1,2,3,4} consist of 4 partitions with 0 blocks of length 1 : {{1,2,3,4}},{{1,2},{3,4}},{{1,3},{2,4}},{{1,4},{2,3}},
%e A086659 4 partitions with 1 block of length 1 : {{1},{2,3,4}},{{1,2,3},{4}},{{1,2,4},{3}},{{1,3,4},{2}}
%e A086659 6 partitions with 2 blocks of length 1 : {{1},{2},{3,4}},{{1},{2,3},{4}},{{1},{2,4},{3}},{{1,2},{3},{4}},{{1,3},{2},{4}},{{1,4},{2},{3}}.
%e A086659 (There are no partitions with n-1 blocks of length 1 and 1 with n of them)
%e A086659     1;
%e A086659     1,   3;
%e A086659     4,   4,   6;
%e A086659    11,  20,  10,  10;
%e A086659    41,  66,  60,  20, 15;
%e A086659   162, 287, 231, 140, 35, 21;
%e A086659   ...
%p A086659 with(combinat):
%p A086659 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A086659       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A086659       b(n-i*j, i-1)*`if`(i=1, x^j, 1), j=0..n/i))))
%p A086659     end:
%p A086659 T:= n-> (p-> seq(coeff(p, x, i), i=0..n-2))(b(n$2)):
%p A086659 seq(T(n), n=2..16);  # _Alois P. Heinz_, Mar 08 2015
%t A086659 Table[Count[Count[ #, {_Integer}]&/@SetPartitions[n], # ]&/@Range[0, n-2], {n, 2, 10}]
%Y A086659 Row sums = Bell[n]-1 (A058692), first column=A000296, main diagonal = triangular numbers A000217.
%K A086659 nonn,tabl,easy
%O A086659 2,3
%A A086659 _Wouter Meeussen_, Jul 27 2003
%E A086659 More terms from _Vladeta Jovovic_, Jul 28 2003