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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 (list; table; graph; refs; listen; history; text; internal format)
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}]

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.

Cf. A001970, A055884, A096751, A144150, A196545, A281113, A289501, A290353, A300383, A323719, A327618, A327639.

Sequence in context: A133815 A305027 A335570 * A130580 A220708 A110541

Adjacent sequences:  A323715 A323716 A323717 * A323719 A323720 A323721

KEYWORD

nonn,tabl

AUTHOR

Gus Wiseman, Jan 25 2019

STATUS

approved

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Last modified April 17 08:34 EDT 2021. Contains 343064 sequences. (Running on oeis4.)