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A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.

%I A111246 #27 Jan 17 2025 19:21:45
%S A111246 1,1,1,1,3,1,0,7,6,1,0,10,25,10,1,0,10,75,65,15,1,0,0,175,315,140,21,
%T A111246 1,0,0,280,1225,980,266,28,1,0,0,280,3780,5565,2520,462,36,1,0,0,0,
%U A111246 9100,26145,19425,5670,750,45,1,0,0,0,15400,102025,125895,56595,11550,1155
%N A111246 Triangle read by rows: a(n,k) = number of partitions of an n-set into exactly k nonempty subsets, each of size <= 3.
%C A111246 a(n,k) = 0 if k > n; a(n,k) = 0 if n > 0 and k < 0; a(n,k) can be extended to negative n and k, just as the Stirling numbers or Pascal's triangle can be extended. The present triangle is called the tri-restricted Stirling numbers of the second kind.
%C A111246 Also the Bell transform of the sequence "a(n) = 1 if n<3 else 0". For the definition of the Bell transform see A264428. - _Peter Luschny_, Jan 27 2016
%D A111246 J. Y. Choi and J. D. H. Smith, On the combinatorics of multi-restricted numbers, Ars. Com., 75(2005), pp. 44-63.
%H A111246 J. Y. Choi and J. D. H. Smith, <a href="https://dx.doi.org/10.1016/j.jcta.2005.10.001">The Tri-restricted Numbers and Powers of Permutation Representations</a>, J. Comb. Math. Comb. Comp. 42 (2002), 113-125.
%H A111246 J. Y. Choi and J. D. H. Smith, <a href="https://dx.doi.org/10.1016/S0012-365X(02)00549-6">On the Unimodality and Combinatorics of the Bessel Numbers</a>, Discrete Math., 264 (2003), 45-53.
%H A111246 J. Y. Choi et al., <a href="https://dx.doi.org/10.1016/j.jcta.2005.10.001">Reciprocity for multirestricted Stirling numbers</a>, J. Combin. Theory 113 A (2006), 1050-1060.
%F A111246 a(n, k) = a(n-1, k-1) + k*a(n-1, k) - binomial(n-1, 3)*a(n-4, k-1).
%F A111246 G.f. = Sum_{k_1+k_2+k_3=k, k_1+ 2k_2+3k_3=n} frac{n!}{(1!)^{k_1}(2!)^{k_2}(3!)^{k_3}k_1!k_2!k_3!}.
%F A111246 E.g.f.: exp(y*(x+x^2/2+x^3/6)). - _Vladeta Jovovic_, Nov 01 2005
%e A111246 a(1,1)=1;
%e A111246 a(2,1)=1; a(2,2)=1;
%e A111246 a(3,1)=1; a(3,2)=3; a(3,3)=1;
%e A111246 a(4,1)=0; a(4,2)=7; a(4,3)=6; a(4,4)=1;
%e A111246 a(5,1)=0; a(5,2)=10; a(5,3)=25; a(5,4)=10; a(5,5)=1;
%e A111246 a(6,1)=0; a(6,2)=10; a(6,3)=75; a(6,4)=65; a(6,5)=15; a(6,6)=1; ...
%p A111246 # The function BellMatrix is defined in A264428.
%p A111246 # Adds (1,0,0,0,...) as column 0.
%p A111246 BellMatrix(n -> `if`(n<3,1,0), 10); # _Peter Luschny_, Jan 27 2016
%t A111246 BellMatrix[f_Function, len_] := With[{t = Array[f, len, 0]}, Table[BellY[n, k, t], {n, 0, len - 1}, {k, 0, len - 1}]];
%t A111246 rows = 12;
%t A111246 M = BellMatrix[If[# < 3, 1, 0]&, rows];
%t A111246 Table[M[[n, k]], {n, 2, rows}, {k, 2, n}] // Flatten (* _Jean-François Alcover_, Jun 24 2018, after _Peter Luschny_ *)
%o A111246 (PARI) row(n) = {x='x+O('x^(n+1)); polcoeff(serlaplace(exp(y*(x+x^2/2+x^3/6))), n, 'x); }
%o A111246 tabl(nn) = for(n=1, nn, print(Vecrev(row(n)/y))) \\ _Jinyuan Wang_, Dec 21 2019
%Y A111246 A144385 and A144402 are other versions of this same triangle.
%Y A111246 Cf. A001680, A008277 (Stirling numbers).
%K A111246 nonn,tabl
%O A111246 1,5
%A A111246 _Ji Young Choi_, Oct 31 2005
%E A111246 More terms from _Vladeta Jovovic_, Nov 01 2005
%E A111246 Recurrence, offset and example corrected by _David Applegate_, Jan 16 2009