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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.

%I A102430 #13 Oct 12 2019 21:14:55
%S A102430 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,
%T A102430 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,
%U A102430 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
%N A102430 Triangle read by rows where T(n,k) is the number of integer partitions of n > 1 into powers of k > 1.
%C A102430 All entries above main diagonal are = 1.
%H A102430 Alois P. Heinz, <a href="/A102430/b102430.txt">Rows n = 2..500, flattened</a>
%F A102430 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)
%e A102430 The T(9,3)=5 partitions of 9 into powers of 3: 111111111, 1111113, 11133, 333, 9.
%e A102430 From _Gus Wiseman_, Jun 07 2019: (Start)
%e A102430 Triangle begins:
%e A102430    2
%e A102430    2  2
%e A102430    4  2  2
%e A102430    4  2  2  2
%e A102430    6  3  2  2  2
%e A102430    6  3  2  2  2  2
%e A102430   10  3  3  2  2  2  2
%e A102430   10  5  3  2  2  2  2  2
%e A102430   14  5  3  3  2  2  2  2  2
%e A102430   14  5  3  3  2  2  2  2  2  2
%e A102430   20  7  4  3  3  2  2  2  2  2  2
%e A102430   20  7  4  3  3  2  2  2  2  2  2  2
%e A102430   26  7  4  3  3  3  2  2  2  2  2  2  2
%e A102430   26  9  4  4  3  3  2  2  2  2  2  2  2  2
%e A102430   36  9  6  4  3  3  3  2  2  2  2  2  2  2  2
%e A102430   36  9  6  4  3  3  3  2  2  2  2  2  2  2  2  2
%e A102430   46 12  6  4  4  3  3  3  2  2  2  2  2  2  2  2  2
%e A102430 Row n = 8 counts the following partitions:
%e A102430   8          3311       44         5111       611        71         8
%e A102430   44         311111     41111      11111111   11111111   11111111   11111111
%e A102430   422        11111111   11111111
%e A102430   2222
%e A102430   4211
%e A102430   22211
%e A102430   41111
%e A102430   221111
%e A102430   2111111
%e A102430   11111111
%e A102430 (End)
%p A102430 b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<0, 0,
%p A102430       b(n, i-1, k)+(p-> `if`(p>n, 0, b(n-p, i, k)))(k^i)))
%p A102430     end:
%p A102430 T:= (n, k)-> b(n, ilog[k](n), k):
%p A102430 seq(seq(T(n, k), k=2..n), n=2..20);  # _Alois P. Heinz_, Oct 12 2019
%t A102430 Table[Length[Select[IntegerPartitions[n],And@@(IntegerQ[Log[k,#]]&/@#)&]],{n,2,10},{k,2,n}] (* _Gus Wiseman_, Jun 07 2019 *)
%Y A102430 Cf. A102431, A102432, A102433, A102434, A001700, A018819, A062051, A008645, A008650, A008652, A008648.
%Y A102430 Same as A308558 except for the k = 1 column.
%Y A102430 Row sums are A102431.
%Y A102430 First column (k = 2) is A018819.
%Y A102430 Second column (k = 3) is A062051.
%Y A102430 Cf. A000961, A001597, A007916, A023894, A052410, A112344.
%K A102430 easy,nonn,tabl
%O A102430 2,1
%A A102430 _Marc LeBrun_, Jan 08 2005
%E A102430 Corrected and rewritten by _Gus Wiseman_, Jun 07 2019