OFFSET
0,8
COMMENTS
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.
LINKS
Alois P. Heinz, Rows n = 0..140, flattened
FORMULA
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).
A(n,k) = Sum_{i=0..k} binomial(k,i) * A327639(n,i). - Alois P. Heinz, Sep 20 2019
EXAMPLE
Array begins:
k=0: k=1: k=2: k=3: k=4: k=5:
n=0: 1 1 1 1 1 1
n=1: 1 1 1 1 1 1
n=2: 1 2 3 4 5 6
n=3: 1 3 6 10 15 21
n=4: 1 5 15 34 65 111
n=5: 1 7 28 80 185 371
n=6: 1 11 66 254 739 1785
n=7: 1 15 122 604 2163 6223
n=8: 1 22 266 1785 8120 28413
n=9: 1 30 503 4370 24446 101534
The A(4,2) = 15 twice-partitions:
(4) (31) (22) (211) (1111)
(3)(1) (2)(2) (11)(2) (11)(11)
(2)(11) (111)(1)
(21)(1) (11)(1)(1)
(2)(1)(1) (1)(1)(1)(1)
MAPLE
b:= proc(n, i, k) option remember; `if`(n=0 or k=0 or i=1,
1, b(n, i-1, k)+b(i$2, k-1)*b(n-i, min(n-i, i), k))
end:
A:= (n, k)-> b(n$2, k):
seq(seq(A(d-k, k), k=0..d), d=0..14); # Alois P. Heinz, Jan 25 2019
MATHEMATICA
ptnlev[n_, k_]:=Switch[k, 0, {n}, 1, IntegerPartitions[n], _, Join@@Table[Tuples[ptnlev[#, k-1]&/@ptn], {ptn, IntegerPartitions[n]}]];
Table[Length[ptnlev[sum-k, k]], {sum, 0, 12}, {k, 0, sum}]
(* Second program: *)
b[n_, i_, k_] := b[n, i, k] = If[n == 0 || k == 0 || i == 1, 1,
b[n, i - 1, k] + b[i, i, k - 1]*b[n - i, Min[n - i, i], k]];
A[n_, k_] := b[n, n, k];
Table[Table[A[d - k, k], {k, 0, d}], {d, 0, 14}] // Flatten (* Jean-François Alcover, May 13 2021, after Alois P. Heinz *)
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 25 2019
STATUS
approved