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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. 7
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois 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 A240694 A258643

Adjacent sequences:  A168014 A168015 A168016 * A168018 A168019 A168020

KEYWORD

nonn,look,tabf

AUTHOR

Omar E. Pol, Nov 22 2009

STATUS

approved

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Last modified October 15 23:54 EDT 2019. Contains 328038 sequences. (Running on oeis4.)