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A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).

%I A124327 #22 Jun 10 2024 10:02:54
%S A124327 1,1,0,1,1,0,2,1,0,1,1,0,4,2,1,3,2,1,0,1,1,0,8,4,2,10,6,7,2,5,3,2,1,0,
%T A124327 1,1,0,16,8,4,29,19,21,14,23,14,18,10,7,7,5,3,2,1,0,1,1,0,32,16,8,85,
%U A124327 56,64,42,101,62,75,69,47,54,38,38,24,23,10,13,7,5,3,2,1,0,1,1,0,64,32,16
%N A124327 Triangle read by rows: T(n,k) is the number of partitions of the set {1,2,...,n} such that the sum of the least entries of the blocks is k (1<=k<=n*(n+1)/2).
%C A124327 Row n has n(n+1)/2 terms. Row sums yield the Bell numbers (A000110). T(n,1)=1; T(n,2)=0; T(n,3)=2^(n-2). for n>=2; T(n,4)=2^(n-3) for n>=3; T(n,5)=2^(n-4) for n>=4.
%H A124327 Alois P. Heinz, <a href="/A124327/b124327.txt">Rows n = 1..40, flattened</a>
%F A124327 The generating polynomial of row n is P(n,t)=Q(n,t,1), where Q(n,t,s)=s*dQ(n-1,t,s)/ds + st^n*Q(n-1,t,s); Q(1,t,s)=ts.
%F A124327 Sum_{k=1..n*(n+1)/2} k * T(n,k) = A124325(n+1). - _Alois P. Heinz_, Dec 05 2023
%e A124327 T(4,7) = 2 because we have 13|2|4 and 1|23|4.
%e A124327 Triangle starts:
%e A124327   1;
%e A124327   1, 0,  1;
%e A124327   1, 0,  2, 1, 0,  1;
%e A124327   1, 0,  4, 2, 1,  3,  2,  1,  0,  1;
%e A124327   1, 0,  8, 4, 2, 10,  6,  7,  2,  5,  3,  2,  1, 0, 1;
%e A124327   1, 0, 16, 8, 4, 29, 19, 21, 14, 23, 14, 18, 10, 7, 7, 5, 3, 2, 1, 0, 1;
%e A124327   ...
%p A124327 Q[1]:=t*s: for n from 2 to 8 do Q[n]:=expand(s*diff(Q[n-1],s)+t^n*s*Q[n-1]) od: for n from 1 to 8 do P[n]:=sort(subs(s=1,Q[n])) od: for n from 1 to 8 do seq(coeff(P[n],t,k),k=1..n*(n+1)/2) od; # yields sequence in triangular form
%t A124327 Q[1, t_, s_] := t s;
%t A124327 Q[n_, t_, s_] := Q[n, t, s] = s D[Q[n-1, t, s], s] + s t^n Q[n-1, t, s] // Expand;
%t A124327 P[n_, t_] := Q[n, t, s] /. s -> 1;
%t A124327 T[n_] := Rest@CoefficientList[P[n, t], t];
%t A124327 Table[T[n], {n, 1, 8}] // Flatten (* _Jean-François Alcover_, Jun 10 2024 *)
%Y A124327 Cf. A000110, A000217, A124325, A367955.
%Y A124327 Antidiagonal sums give A365821.
%Y A124327 Row maxima give A365903.
%Y A124327 T(n,n) gives A368204.
%K A124327 nonn,tabf
%O A124327 1,7
%A A124327 _Emeric Deutsch_, Oct 31 2006