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A059022 Triangle of Stirling numbers of order 3.

%I A059022 #39 Apr 30 2024 10:18:53
%S A059022 1,1,1,1,10,1,35,1,91,1,210,280,1,456,2100,1,957,10395,1,1969,42735,
%T A059022 15400,1,4004,158301,200200,1,8086,549549,1611610,1,16263,1827826,
%U A059022 10335325,1401400,1,32631,5903898,57962905,28028000,1,65382,18682014,297797500
%N A059022 Triangle of Stirling numbers of order 3.
%C A059022 The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 3. This is the 3-associated Stirling number of the second kind (Comtet) or the Stirling number of order 3 (Fekete).
%C A059022 This is entered as a triangular array. The entries S_3(n,k) are zero for 3k>n, so these values are omitted. The initial entry in the sequence is S_3(3,1).
%C A059022 Rows are of lengths 1,1,1,2,2,2,3,3,3,...
%D A059022 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
%D A059022 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
%H A059022 Gilles Bonnet and Anna Gusakova, <a href="https://arxiv.org/abs/2404.16756">Concentration inequalities for Poisson U-statistics</a>, arXiv:2404.16756 [math.PR], 2024. See p. 17.
%H A059022 Antal E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%H A059022 Gergő Nemes, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL13/Nemes/nemes2.html">On the Coefficients of the Asymptotic Expansion of n!</a>, J. Int. Seq. 13 (2010), 10.6.6.
%F A059022 S_r(n+1,k) = k*S_r(n,k) + binomial(n,r-1)*S_r(n-r+1,k-1); for this sequence, r=3.
%F A059022 G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t - Sum_{i=0..r-1} t^i/i!)).
%F A059022 T(n,k) = Sum_{j=0..min(n/2,k)} (-1)^j*B(n,j)*S_2(n-2j,k-j), where B are the Bessel numbers A100861 and S_2 are the 2-associated Stirling numbers of the second kind A008299. - _Fabián Pereyra_, Feb 20 2022
%e A059022 There are 10 ways of partitioning a set N of cardinality 6 into 2 blocks each of cardinality at least 3, so S_3(6,2) = 10.
%e A059022 From _Wesley Ivan Hurt_, Feb 24 2022: (Start)
%e A059022 Triangle starts:
%e A059022   1;
%e A059022   1;
%e A059022   1;
%e A059022   1,   10;
%e A059022   1,   35;
%e A059022   1,   91;
%e A059022   1,  210,   280;
%e A059022   1,  456,  2100;
%e A059022   1,  957, 10395;
%e A059022   1, 1969, 42735, 15400;
%e A059022   ...
%e A059022 (End)
%p A059022 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A059022       expand(x*b(n-j))*binomial(n-1, j-1), j=3..n))
%p A059022     end:
%p A059022 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A059022 seq(T(n), n=3..20);  # _Alois P. Heinz_, Feb 21 2022
%p A059022 # alternative
%p A059022 A059022 := proc(n, k)
%p A059022     option remember;
%p A059022     if n<3 then
%p A059022         0;
%p A059022     elif n < 6 and k=1 then
%p A059022         1 ;
%p A059022     else
%p A059022         k*procname(n-1, k)+binomial(n-1, 2)*procname(n-3, k-1) ;
%p A059022     end if;
%p A059022 end proc:  # _R. J. Mathar_, Apr 15 2022
%t A059022 S3[3, 1] = S3[4, 1] = S3[5, 1] = 1; S3[n_, k_] /; 1 <= k <= Floor[n/3] := S3[n, k] = k*S3[n-1, k] + Binomial[n-1, 2]*S3[n-3, k-1]; S3[_, _] = 0; Flatten[ Table[ S3[n, k], {n, 3, 20}, {k, 1, Floor[n/3]}]] (* _Jean-François Alcover_, Feb 21 2012 *)
%Y A059022 Row sums give A006505.
%Y A059022 Cf. A008299, A059023, A059024, A059025, A100861, A272352 (column 2), A272982 (column 3), A261724 (column 4), A352611 (column 5).
%K A059022 nonn,tabf,nice
%O A059022 3,5
%A A059022 Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000