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A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).

%I A124503 #23 Aug 04 2025 05:36:15
%S A124503 1,1,2,4,1,11,4,32,20,113,80,10,422,385,70,1788,1792,560,8015,9492,
%T A124503 3360,280,39435,50640,23100,2800,204910,295020,147840,30800,1144377,
%U A124503 1763300,1044120,246400,15400,6722107,11278410,7241520,2202200,200200,41877722
%N A124503 Triangle read by rows: T(n,k) is the number of set partitions of the set {1,2,...,n} (or of any n-set) containing k blocks of size 3 (0<=k<=floor(n/3)).
%C A124503 Row n contains 1+floor(n/3) terms. Row sums yield the Bell numbers (A000110). T(n,0)=A124504(n). Sum(k*T(n,k), k=0..floor(n/3))=A105480(n+1).
%H A124503 Alois P. Heinz, <a href="/A124503/b124503.txt">Rows n = 0..250, flattened</a>
%F A124503 E.g.f.: G(t,z) = exp(exp(z)-1+(t-1)z^3/6).
%e A124503 T(4,1)=4 because we have 1|234, 134|2, 124|3 and 123|4.
%e A124503 Triangle starts:
%e A124503     1;
%e A124503     1;
%e A124503     2;
%e A124503     4,   1;
%e A124503    11,   4;
%e A124503    32,  20;
%e A124503   113,  80, 10;
%e A124503   422, 385, 70;
%e A124503   ...
%p A124503 G:=exp(exp(z)-1+(t-1)*z^3/6): Gser:=simplify(series(G,z=0,17)): for n from 0 to 14 do P[n]:=sort(n!*coeff(Gser,z,n)) od: for n from 0 to 14 do seq(coeff(P[n],t,k),k=0..floor(n/3)) od; # yields sequence in triangular form
%p A124503 # second Maple program:
%p A124503 with(combinat):
%p A124503 b:= proc(n, i) option remember; expand(`if`(n=0, 1,
%p A124503       `if`(i<1, 0, add(multinomial(n, n-i*j, i$j)/j!*
%p A124503       b(n-i*j, i-1)*`if`(i=3, x^j, 1), j=0..n/i))))
%p A124503     end:
%p A124503 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n$2)):
%p A124503 seq(T(n), n=0..15);  # _Alois P. Heinz_, Mar 08 2015
%t A124503 nn = 8; k = 3; Range[0, nn]! CoefficientList[Series[Exp[Exp[x] - 1 + (y - 1) x^k/k!], {x, 0, nn}], {x, y}] // Grid (* _Geoffrey Critzer_, Aug 26 2012 *)
%Y A124503 Cf. A000110, A124504, A105480, A355144.
%Y A124503 T(3n,n) gives A025035.
%K A124503 nonn,tabf
%O A124503 0,3
%A A124503 _Emeric Deutsch_, Nov 14 2006