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A327632 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.

%I A327632 #26 Dec 09 2020 15:14:20
%S A327632 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 A327632 7,1,17,121,384,638,548,196,8,1,24,210,880,1983,2442,1543,400,9,1,29,
%U A327632 353,2012,6211,10865,10555,5231,1026,10,1,39,600,4477,17883,40855,54279,40511,15178,2070,11
%N A327632 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 A327632 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 A327632 T(n,k) is defined for all n>=0 and k>=0.  The triangle displays only positive terms.  All other terms are zero.
%C A327632 Row n is the inverse binomial transform of the n-th row of array A327622.
%H A327632 Alois P. Heinz, <a href="/A327632/b327632.txt">Rows n = 1..200, flattened</a>
%H A327632 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_(number_theory)">Partition (number theory)</a>
%F A327632 T(n,k) = Sum_{i=0..k} (-1)^(k-i) * binomial(k,i) * A327622(n,i).
%F A327632 T(n+1,n-1) = 1 for n >= 1.
%e A327632 T(4,0) = 1:
%e A327632   4    (1 part).
%e A327632 T(4,1) = 2:
%e A327632   4-> 31    (2 parts)
%e A327632 T(4,2) = 3:
%e A327632   4-> 31 -> 211   (3 parts)
%e A327632 Triangle T(n,k) begins:
%e A327632   1;
%e A327632   1;
%e A327632   1,  2;
%e A327632   1,  2,   3;
%e A327632   1,  4,   6,    4;
%e A327632   1,  7,  13,   12,    5;
%e A327632   1,  9,  30,   52,   35,     6;
%e A327632   1, 12,  61,  137,  156,    72,     7;
%e A327632   1, 17, 121,  384,  638,   548,   196,    8;
%e A327632   1, 24, 210,  880, 1983,  2442,  1543,  400,    9;
%e A327632   1, 29, 353, 2012, 6211, 10865, 10555, 5231, 1026, 10;
%e A327632   ...
%p A327632 b:= proc(n, i, k) option remember; `if`(n=0, [1, 0],
%p A327632      `if`(k=0, [1, 1], `if`(i*(i+1)/2<n, 0, b(n, i-1, k)+
%p A327632          (h-> (f-> f +[0, f[1]*h[2]/h[1]])(h[1]*
%p A327632         b(n-i, min(n-i, i-1), k)))(b(i$2, k-1)))))
%p A327632     end:
%p A327632 T:= (n, k)-> add(b(n$2, i)[2]*(-1)^(k-i)*binomial(k, i), i=0..k):
%p A327632 seq(seq(T(n, k), k=0..max(0, n-2)), n=1..14);
%t A327632 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 A327632 T[n_, k_] := Sum[b[n, n, i][[2]] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t A327632 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 A327632 Columns k=0-2 give: A057427, -1+A015723(n), A327795.
%Y A327632 Row sums give A327647.
%Y A327632 Cf. A327622, A327631.
%K A327632 nonn,tabf
%O A327632 1,4
%A A327632 _Alois P. Heinz_, Sep 19 2019