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%I #41 Dec 09 2020 15:12:15
%S 1,1,1,1,1,2,1,1,4,6,3,1,6,15,16,6,1,10,45,88,76,24,1,14,93,282,420,
%T 302,84,1,21,223,1052,2489,3112,1970,498,1,29,444,2950,9865,18123,
%U 18618,10046,2220,1,41,944,9030,42787,112669,173338,155160,74938,15108
%N Number T(n,k) of proper k-times partitions of n; triangle T(n,k), n >= 0, 0 <= k <= max(0,n-1), read by rows.
%C In each step at least one part is replaced by the partition of itself into smaller parts. The parts are not resorted.
%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 A323718.
%H Alois P. Heinz, <a href="/A327639/b327639.txt">Rows n = 0..170, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Iverson_bracket">Iverson bracket</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) * A323718(n,i).
%F T(n,n-1) = A327631(n,n-1)/n = A327643(n) for n >= 1.
%F Sum_{k=1..n-1} k * T(n,k) = A327646(n).
%F Sum_{k=0..max(0,n-1)} (-1)^k * T(n,k) = [n<2], where [] is an Iverson bracket.
%e T(4,0) = 1: 4
%e T(4,1) = 4: T(4,2) = 6: T(4,3) = 3:
%e 4-> 31 4-> 31 -> 211 4-> 31 -> 211 -> 1111
%e 4-> 22 4-> 31 -> 1111 4-> 22 -> 112 -> 1111
%e 4-> 211 4-> 22 -> 112 4-> 22 -> 211 -> 1111
%e 4-> 1111 4-> 22 -> 211
%e 4-> 22 -> 1111
%e 4-> 211-> 1111
%e Triangle T(n,k) begins:
%e 1;
%e 1;
%e 1, 1;
%e 1, 2, 1;
%e 1, 4, 6, 3;
%e 1, 6, 15, 16, 6;
%e 1, 10, 45, 88, 76, 24;
%e 1, 14, 93, 282, 420, 302, 84;
%e 1, 21, 223, 1052, 2489, 3112, 1970, 498;
%e 1, 29, 444, 2950, 9865, 18123, 18618, 10046, 2220;
%e 1, 41, 944, 9030, 42787, 112669, 173338, 155160, 74938, 15108;
%e ...
%p b:= proc(n, i, k) option remember; `if`(n=0 or k=0, 1, `if`(i>1,
%p b(n, i-1, k), 0) +b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p end:
%p T:= (n, k)-> add(b(n$2, i)*(-1)^(k-i)*binomial(k, i), i=0..k):
%p seq(seq(T(n, k), k=0..max(0, n-1)), n=0..12);
%t b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0, 1, If[i > 1, b[n, i - 1, k], 0] + b[i, i, k - 1] b[n - i, Min[n - i, i], k]];
%t T[n_, k_] := Sum[b[n, n, i] (-1)^(k - i) Binomial[k, i], {i, 0, k}];
%t Table[T[n, k], {n, 0, 12}, {k, 0, Max[0, n - 1] }] // Flatten (* _Jean-François Alcover_, Dec 09 2020, after _Alois P. Heinz_ *)
%Y Columns k=0-2 give A000012, A000065, A327769.
%Y Row sums give A327644.
%Y T(2n,n) gives A327645.
%Y Cf. A323718, A327631, A327643, A327646.
%K nonn,tabf
%O 0,6
%A _Alois P. Heinz_, Sep 20 2019
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