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Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.
1

%I #5 Jun 08 2019 17:55:06

%S 1,1,2,1,2,2,1,4,2,2,1,4,2,2,2,1,6,3,2,2,2,1,6,3,2,2,2,2,1,10,3,3,2,2,

%T 2,2,1,10,5,3,2,2,2,2,2,1,14,5,3,3,2,2,2,2,2,1,14,5,3,3,2,2,2,2,2,2,1,

%U 20,7,4,3,3,2,2,2,2,2,2,1,20,7,4,3,3,2

%N Triangle read by rows where T(n,k) is the number of integer partitions of n > 0 into powers of k > 0.

%e Triangle begins:

%e 1

%e 1 2

%e 1 2 2

%e 1 4 2 2

%e 1 4 2 2 2

%e 1 6 3 2 2 2

%e 1 6 3 2 2 2 2

%e 1 10 3 3 2 2 2 2

%e 1 10 5 3 2 2 2 2 2

%e 1 14 5 3 3 2 2 2 2 2

%e 1 14 5 3 3 2 2 2 2 2 2

%e 1 20 7 4 3 3 2 2 2 2 2 2

%e 1 20 7 4 3 3 2 2 2 2 2 2 2

%e Row n = 6 counts the following partitions:

%e (111111) (42) (33) (411) (51) (6)

%e (222) (3111) (111111) (111111) (111111)

%e (411) (111111)

%e (2211)

%e (21111)

%e (111111)

%t Table[If[k==1,1,Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]]],{n,10},{k,n}]

%Y Same as A102430 except for the k = 1 column.

%Y Row sums are A102431(n) + 1.

%Y Column k = 2 is A018819.

%Y Column k = 3 is A062051.

%Y Cf. A000961, A001597, A007916, A008284, A023894, A052410, A101417, A112344, A323053, A323054.

%K nonn,tabl

%O 1,3

%A _Gus Wiseman_, Jun 07 2019