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A059023 Triangle of Stirling numbers of order 4.

%I A059023 #28 Apr 15 2022 08:05:54
%S A059023 1,1,1,1,1,35,1,126,1,336,1,792,1,1749,5775,1,3718,45045,1,7722,
%T A059023 231231,1,15808,981981,1,32071,3741738,2627625,1,64702,13307294,
%U A059023 35735700,1,130084,45172842,300179880,1,260984,148417854,2002016016,1,522937,476330361
%N A059023 Triangle of Stirling numbers of order 4.
%C A059023 The number of partitions of the set N, |N|=n, into k blocks, all of cardinality greater than or equal to 4. This is the 4-associated Stirling number of the second kind.
%C A059023 This is entered as a triangular array. The entries S_4(n,k) are zero for 4k>n, so these values are omitted. Initial entry in sequence is S_4(4,1).
%C A059023 Rows are of lengths 1,1,1,1,2,2,2,2,3,3,3,3,...
%D A059023 L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 222.
%D A059023 J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 76.
%H A059023 Alois P. Heinz, <a href="/A059023/b059023.txt">Rows n = 4..300, flattened</a>
%H A059023 A. E. Fekete, <a href="http://www.jstor.org/stable/2974533">Apropos two notes on notation</a>, Amer. Math. Monthly, 101 (1994), 771-778.
%F A059023 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=4.
%F A059023 G.f.: Sum_{n>=0, k>=0} S_r(n,k)*u^k*t^n/n! = exp(u(e^t-sum(t^i/i!, i=0..r-1))).
%F A059023 T(n,k) = Sum_{j=0..min(n/3,k)} (-1)^j*n!/(6^j*j!*(n-3j)!)*S_3(n-3j,k-j), where S_3 are the 3-associated Stirling numbers of the second kind A059022. - _Fabián Pereyra_, Feb 21 2022
%e A059023 There are 35 ways of partitioning a set N of cardinality 8 into 2 blocks each of cardinality at least 4, so S_4(8,2) = 35.
%p A059023 b:= proc(n) option remember; `if`(n=0, 1, add(
%p A059023       expand(x*b(n-j))*binomial(n-1, j-1), j=4..n))
%p A059023     end:
%p A059023 T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n)):
%p A059023 seq(T(n), n=4..20);  # _Alois P. Heinz_, Feb 21 2022
%p A059023 # alternative
%p A059023 A059023 := proc(n, k)
%p A059023     option remember;
%p A059023     if n<4 then
%p A059023         0;
%p A059023     elif n < 8 and k=1 then
%p A059023         1 ;
%p A059023     else
%p A059023         k*procname(n-1, k)+binomial(n-1, 3)*procname(n-4, k-1) ;
%p A059023     end if;
%p A059023 end proc:  # _R. J. Mathar_, Apr 15 2022
%t A059023 s4[n_, k_] := k*s4[n-1, k] + Binomial[n-1, 3]*s4[n-4, k-1]; s4[n_, k_] /; 4 k > n = 0; s4[_, k_ /; k <= 0] = 0; s4[0, 0] = 1;
%t A059023 Flatten[Table[s4[n, k], {n, 4, 20}, {k, 1, Floor[n/4]}]][[1 ;; 42]] (* _Jean-François Alcover_, Jun 16 2011 *)
%Y A059023 Row sums give A057837.
%Y A059023 Cf. A008299, A059022, A059024, A059025.
%K A059023 nonn,tabf,nice
%O A059023 4,6
%A A059023 Barbara Haas Margolius (margolius(AT)math.csuohio.edu), Dec 14 2000