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.

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, 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, 385, 1, 490, 1, 2, 5, 7, 42, 627, 1, 3, 15, 792, 1, 2, 56, 1002

Offset

1,3

Comments

Positive values of triangle A168016.

The number of terms of row n is equal to the number of divisors of n: A000005(n).

Note that the last term of each row is the number of partitions of n: A000041(n).

Also, it appears that row n lists the partition numbers of the divisors of n. [Omar E. Pol, Nov 23 2009]

Links

Alois P. Heinz, Rows n = 1..1400, flattened

Omar E. Pol, Illustration of the partitions of n, for n = 1 .. 9

Example

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.
Triangle begins:
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,  3,  5, 11, 77;

Maple

with(numtheory):
b:= proc(n, i, d) option remember;
      if n<0 then 0
    elif n=0 then 1
    elif i<1 then 0
    else b(n, i-d, d) +b(n-i, i, d)
      fi
    end:
T:= proc(n) local l;
      l:= sort([divisors(n)[]], `>`);
      seq(b(n, n, l[i]), i=1..nops(l))
    end:
seq(T(n), n=1..30); # Alois P. Heinz, Oct 21 2011

Mathematica

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 *)

Crossrefs

Row sums give A047968.

Cf. A000005, A000041, A056538, A135010, A138121, A168016, A168018, A168019, A168020, A168021.

Sequence in context: A080521 A169613 A176572 * A293980 A365785 A368759

Adjacent sequences: A168014 A168015 A168016 * A168018 A168019 A168020

Keyword

nonn,look,tabf

Author

Omar E. Pol, Nov 22 2009

Status

approved