%I #26 Dec 09 2020 15:14:20
%S 1,1,1,2,1,2,3,1,4,6,4,1,7,13,12,5,1,9,30,52,35,6,1,12,61,137,156,72,
%T 7,1,17,121,384,638,548,196,8,1,24,210,880,1983,2442,1543,400,9,1,29,
%U 353,2012,6211,10865,10555,5231,1026,10,1,39,600,4477,17883,40855,54279,40511,15178,2070,11
%N Number T(n,k) of parts in all proper k-times partitions of n into distinct parts; triangle T(n,k), n >= 1, 0 <= k <= max(0,n-2), read by rows.
%C In each step at least one part is replaced by the partition of itself into smaller distinct parts. The parts are not resorted and the parts in the result are not necessarily distinct.
%C T(n,k) is defined for all n>=0 and k>=0. The triangle displays only positive terms. All other terms are zero.
%C Row n is the inverse binomial transform of the n-th row of array A327622.
%H Alois P. Heinz, <a href="/A327632/b327632.txt">Rows n = 1..200, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327622(n,i).
%F T(n+1,n-1) = 1 for n >= 1.
%e T(4,0) = 1:
%e 4 (1 part).
%e T(4,1) = 2:
%e 4-> 31 (2 parts)
%e T(4,2) = 3:
%e 4-> 31 -> 211 (3 parts)
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 1, 2;
%e 1, 2, 3;
%e 1, 4, 6, 4;
%e 1, 7, 13, 12, 5;
%e 1, 9, 30, 52, 35, 6;
%e 1, 12, 61, 137, 156, 72, 7;
%e 1, 17, 121, 384, 638, 548, 196, 8;
%e 1, 24, 210, 880, 1983, 2442, 1543, 400, 9;
%e 1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10;
%e ...
%p b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p end:
%p T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..max(0, n-2)), n=1..14);
%t b[n_, i_, k_] := b[n, i, k] = With[{}, If[n == 0, {1, 0}, If[k == 0, {1, 1}, If[i (i + 1)/2 < n, {0, 0}, b[n, i - 1, k] + Function[h, Function[f, f + {0, f[[1]] h[[2]]/h[[1]]}][h[[1]] b[n - i, Min[n - i, i - 1], k]]][ b[i, i, k - 1]]]]]];
%t T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t Table[Table[T[n, k], {k, 0, Max[0, n - 2]}], {n, 1, 14}] // Flatten (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-2 give: A057427, -1+A015723(n), A327795.
%Y Row sums give A327647.
%Y Cf. A327622, A327631.
%K nonn,tabf
%O 1,4
%A _Alois P. Heinz_, Sep 19 2019