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A102430
Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.
12
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
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
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.
Sequence in context: A075016 A279409 A102445 * A160691 A367626 A049716
KEYWORD
easy,nonn,tabl
AUTHOR
Marc LeBrun, Jan 08 2005
EXTENSIONS
Corrected and rewritten by Gus Wiseman, Jun 07 2019
STATUS
approved