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A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.

%I A241500 #13 Jul 28 2021 06:50:34
%S A241500 1,1,1,1,2,1,1,5,4,1,1,11,16,7,1,1,23,58,41,11,1,1,47,196,215,90,16,1,
%T A241500 1,95,634,1041,640,176,22,1,1,191,1996,4767,4151,1631,315,29,1,1,383,
%U A241500 6178,21001,25221,13587,3696,526,37,1,1,767,18916,90055,146140,105042,38409,7638,831,46,1
%N A241500 Triangle T(n,k): number of ways of partitioning the n-element multiset {1,1,2,3,...,n-1} into exactly k nonempty parts, n>=1 and 1<=k<=n.
%F A241500 T(n,k) = S(n-1,k) + S(n-1,k-1) + C(k,2)*S(n-2,k), where S refers to Stirling numbers of the second kind (A008277), and C to binomial coefficients (A007318).
%e A241500 There are 58 ways to partition {1,1,2,3,4,5} into three nonempty parts.
%e A241500 The first few rows are:
%e A241500   1;
%e A241500   1,   1;
%e A241500   1,   2,    1;
%e A241500   1,   5,    4,     1;
%e A241500   1,  11,   16,     7,     1;
%e A241500   1,  23,   58,    41,    11,     1;
%e A241500   1,  47,  196,   215,    90,    16,    1;
%e A241500   1,  95,  634,  1041,   640,   176,   22,   1;
%e A241500   1, 191, 1996,  4767,  4151,  1631,  315,  29,  1;
%e A241500   1, 383, 6178, 21001, 25221, 13587, 3696, 526, 37, 1;
%e A241500   ...
%o A241500 (PARI) T(n,k) = stirling(n-1,k,2) + stirling(n-1,k-1,2) + binomial(k,2)*stirling(n-2,k,2); \\ _Michel Marcus_, Apr 24 2014
%Y A241500 The first five columns appear as A000012, A083329, A168583, A168584, A168585.
%Y A241500 Row sums give A035098.
%K A241500 nonn,easy,tabl
%O A241500 1,5
%A A241500 _Andrew Woods_, Apr 24 2014