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A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.

%I A168017 #22 Apr 02 2017 00:33:36
%S A168017 1,1,2,1,3,1,2,5,1,7,1,2,3,11,1,15,1,2,5,22,1,3,30,1,2,7,42,1,56,1,2,
%T A168017 3,5,11,77,1,101,1,2,15,135,1,3,7,176,1,2,5,22,231,1,297,1,2,3,11,30,
%U A168017 385,1,490,1,2,5,7,42,627,1,3,15,792,1,2,56,1002
%N A168017 Triangle read by rows in which row n lists the number of partitions of n into parts divisible by d, where d is a divisor of n listed in decreasing order.
%C A168017 Positive values of triangle A168016.
%C A168017 The number of terms of row n is equal to the number of divisors of n: A000005(n).
%C A168017 Note that the last term of each row is the number of partitions of n: A000041(n).
%C A168017 Also, it appears that row n lists the partition numbers of the divisors of n. [_Omar E. Pol_, Nov 23 2009]
%H A168017 Alois P. Heinz, <a href="/A168017/b168017.txt">Rows n = 1..1400, flattened</a>
%H A168017 Omar E. Pol, <a href="http://www.polprimos.com/imagenespub/polpa3dt.jpg">Illustration of the partitions of n, for n = 1 .. 9</a>
%e A168017 Consider row n=8: (1, 2, 5, 22). The divisors of 8 listed in decreasing order are 8, 4, 2, 1 (see A056538). There is 1 partition of 8 into parts divisible by 8. Also, there are 2 partitions of 8 into parts divisible by 4: {(8), (4+4)}; 5 partitions of 8 into parts divisible by 2: {(8), (6+2), (4+4), (4+2+2), (2+2+2+2)}; and 22 partitions of 8 into parts divisible by 1, because A000041(8)=22. Then row 8 is formed by 1, 2, 5, 22.
%e A168017 Triangle begins:
%e A168017 1;
%e A168017 1,  2;
%e A168017 1,  3;
%e A168017 1,  2,  5;
%e A168017 1,  7;
%e A168017 1,  2,  3, 11;
%e A168017 1, 15;
%e A168017 1,  2,  5, 22;
%e A168017 1,  3, 30;
%e A168017 1,  2,  7, 42;
%e A168017 1, 56;
%e A168017 1,  2,  3,  5, 11, 77;
%p A168017 with(numtheory):
%p A168017 b:= proc(n, i, d) option remember;
%p A168017       if n<0 then 0
%p A168017     elif n=0 then 1
%p A168017     elif i<1 then 0
%p A168017     else b(n, i-d, d) +b(n-i, i, d)
%p A168017       fi
%p A168017     end:
%p A168017 T:= proc(n) local l;
%p A168017       l:= sort([divisors(n)[]],`>`);
%p A168017       seq(b(n, n, l[i]), i=1..nops(l))
%p A168017     end:
%p A168017 seq(T(n), n=1..30); # _Alois P. Heinz_, Oct 21 2011
%t A168017 b[n_, i_, d_] := b[n, i, d] = Which[n<0, 0, n==0, 1, i<1, 0, True, b[n, i - d, d] + b[n-i, i, d]]; T[n_] := Module[{l = Divisors[n] // Reverse}, Table[b[n, n, l[[i]]], {i, 1, Length[l]}]]; Table[T[n], {n, 1, 30}] // Flatten (* _Jean-François Alcover_, Dec 03 2015, after _Alois P. Heinz_ *)
%Y A168017 Row sums give A047968.
%Y A168017 Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.
%K A168017 nonn,look,tabf
%O A168017 1,3
%A A168017 _Omar E. Pol_, Nov 22 2009