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
Examples
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)
Links
- Alois P. Heinz, Rows n = 2..500, flattened
Crossrefs
Programs
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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 *)
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)
Extensions
Corrected and rewritten by Gus Wiseman, Jun 07 2019
Comments