A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.

2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 6, 3, 2, 2, 2, 6, 3, 2, 2, 2, 2, 10, 3, 3, 2, 2, 2, 2, 10, 5, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 14, 5, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 20, 7, 4, 3, 3, 2, 2, 2, 2, 2, 2, 2, 26, 7, 4, 3, 3, 3, 2, 2, 2, 2, 2, 2, 2

Offset

2,1

Comments

All entries above main diagonal are = 1.

Links

Alois P. Heinz, Rows n = 2..500, flattened

Formula

T(1, k) = 1, T(n, 1) = choose(2n-1, n), T(n>1, k>1) = T(n-1, k) + (T(n/k, k) if k divides n, else 0)

Example

The T(9,3)=5 partitions of 9 into powers of 3: 111111111, 1111113, 11133, 333, 9.
From Gus Wiseman, Jun 07 2019: (Start)
Triangle begins:
   2
   2  2
   4  2  2
   4  2  2  2
   6  3  2  2  2
   6  3  2  2  2  2
  10  3  3  2  2  2  2
  10  5  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2
  14  5  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2
  20  7  4  3  3  2  2  2  2  2  2  2
  26  7  4  3  3  3  2  2  2  2  2  2  2
  26  9  4  4  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2
  36  9  6  4  3  3  3  2  2  2  2  2  2  2  2  2
  46 12  6  4  4  3  3  3  2  2  2  2  2  2  2  2  2
Row n = 8 counts the following partitions:
  8          3311       44         5111       611        71         8
  44         311111     41111      11111111   11111111   11111111   11111111
  422        11111111   11111111
  2222
  4211
  22211
  41111
  221111
  2111111
  11111111
(End)

Maple

b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<0, 0,
      b(n, i-1, k)+(p-> `if`(p>n, 0, b(n-p, i, k)))(k^i)))
    end:
T:= (n, k)-> b(n, ilog[k](n), k):
seq(seq(T(n, k), k=2..n), n=2..20);  # Alois P. Heinz, Oct 12 2019

Mathematica

Table[Length[Select[IntegerPartitions[n], And@@(IntegerQ[Log[k, #]]&/@#)&]], {n, 2, 10}, {k, 2, n}] (* Gus Wiseman, Jun 07 2019 *)

Crossrefs

Cf. A102431, A102432, A102433, A102434, A001700, A018819, A062051, A008645, A008650, A008652, A008648.

Same as A308558 except for the k = 1 column.

Row sums are A102431.

First column (k = 2) is A018819.

Second column (k = 3) is A062051.

Cf. A000961, A001597, A007916, A023894, A052410, A112344.

Sequence in context: A075016 A279409 A102445 * A160691 A367626 A049716

Adjacent sequences: A102427 A102428 A102429 * A102431 A102432 A102433

Keyword

easy,nonn,tabl

Author

Marc LeBrun, Jan 08 2005

Extensions

Corrected and rewritten by Gus Wiseman, Jun 07 2019

Status

approved