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A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.

%I A323718 #31 May 14 2021 08:21:18
%S A323718 1,1,1,1,1,1,1,2,1,1,1,3,3,1,1,1,5,6,4,1,1,1,7,15,10,5,1,1,1,11,28,34,
%T A323718 15,6,1,1,1,15,66,80,65,21,7,1,1,1,22,122,254,185,111,28,8,1,1,1,30,
%U A323718 266,604,739,371,175,36,9,1,1,1,42,503,1785,2163,1785,672,260,45,10,1,1
%N A323718 Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.
%C A323718 A k-times partition of n for k > 1 is a sequence of (k-1)-times partitions, one of each part in an integer partition of n. A 1-times partition of n is just an integer partition of n, and the only 0-times partition of n is the number n itself.
%H A323718 Alois P. Heinz, <a href="/A323718/b323718.txt">Rows n = 0..140, flattened</a>
%F A323718 Column k is the formal power product transform of column k-1, where the formal power product transform of a sequence q with offset 1 is the sequence whose ordinary generating function is Product_{n >= 1} 1/(1 - q(n) * x^n).
%F A323718 A(n,k) = Sum_{i=0..k} binomial(k,i) * A327639(n,i). - _Alois P. Heinz_, Sep 20 2019
%e A323718 Array begins:
%e A323718        k=0:   k=1:   k=2:   k=3:   k=4:   k=5:
%e A323718   n=0:  1      1      1      1      1      1
%e A323718   n=1:  1      1      1      1      1      1
%e A323718   n=2:  1      2      3      4      5      6
%e A323718   n=3:  1      3      6     10     15     21
%e A323718   n=4:  1      5     15     34     65    111
%e A323718   n=5:  1      7     28     80    185    371
%e A323718   n=6:  1     11     66    254    739   1785
%e A323718   n=7:  1     15    122    604   2163   6223
%e A323718   n=8:  1     22    266   1785   8120  28413
%e A323718   n=9:  1     30    503   4370  24446 101534
%e A323718 The A(4,2) = 15 twice-partitions:
%e A323718   (4)  (31)    (22)    (211)      (1111)
%e A323718        (3)(1)  (2)(2)  (11)(2)    (11)(11)
%e A323718                        (2)(11)    (111)(1)
%e A323718                        (21)(1)    (11)(1)(1)
%e A323718                        (2)(1)(1)  (1)(1)(1)(1)
%p A323718 b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
%p A323718       1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
%p A323718     end:
%p A323718 A:= (n, k)-> b(n$2, k):
%p A323718 seq(seq(A(d-k, k), k=0..d), d=0..14);  # _Alois P. Heinz_, Jan 25 2019
%t A323718 ptnlev[n_,k_]:=Switch[k,0,{n},1,IntegerPartitions[n],_,Join@@Table[Tuples[ptnlev[#,k-1]&/@ptn],{ptn,IntegerPartitions[n]}]];
%t A323718 Table[Length[ptnlev[sum-k,k]],{sum,0,12},{k,0,sum}]
%t A323718 (* Second program: *)
%t A323718 b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1,
%t A323718      b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
%t A323718 A[n_, k_] := b[n, n, k];
%t A323718 Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* _Jean-François Alcover_, May 13 2021, after _Alois P. Heinz_ *)
%Y A323718 Columns: A000012 (k=0), A000041 (k=1), A063834 (k=2), A301595 (k=3).
%Y A323718 Rows: A000027 (n=2), A000217 (n=3), A006003 (n=4).
%Y A323718 Main diagonal gives A306187.
%Y A323718 Cf. A001970, A055884, A096751, A144150, A196545, A281113, A289501, A290353, A300383, A323719, A327618, A327639.
%K A323718 nonn,tabl
%O A323718 0,8
%A A323718 _Gus Wiseman_, Jan 25 2019