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A323718
Array read by antidiagonals upwards where A(n,k) is the number of k-times partitions of n.
10
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, 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, 266, 604, 739, 371, 175, 36, 9, 1, 1, 1, 42, 503, 1785, 2163, 1785, 672, 260, 45, 10, 1, 1
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
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
Columns: A000012 (k=0), A000041 (k=1), A063834 (k=2), A301595 (k=3).
Rows: A000027 (n=2), A000217 (n=3), A006003 (n=4).
Main diagonal gives A306187.
Sequence in context: A305027 A335570 A362644 * A130580 A220708 A110541
KEYWORD
nonn,tabl
AUTHOR
Gus Wiseman, Jan 25 2019
STATUS
approved