A221530 Triangle read by rows: T(n,k) = A000005(k)*A000041(n-k).

1, 1, 2, 2, 2, 2, 3, 4, 2, 3, 5, 6, 4, 3, 2, 7, 10, 6, 6, 2, 4, 11, 14, 10, 9, 4, 4, 2, 15, 22, 14, 15, 6, 8, 2, 4, 22, 30, 22, 21, 10, 12, 4, 4, 3, 30, 44, 30, 33, 14, 20, 6, 8, 3, 4, 42, 60, 44, 45, 22, 28, 10, 12, 6, 4, 2, 56, 84, 60, 66, 30, 44, 14, 20, 9, 8, 2, 6

Offset: 1

Author

Omar E. Pol, Jan 19 2013

Keywords

• nonn
• tabl

Comments

T(n,k) is the number of partitions of n that contain k as a part multiplied by the number of divisors of k.

It appears that T(n,k) is also the total number of appearances of k in the last k sections of the set of partitions of n multiplied by the number of divisors of k.

T(n,k) is also the number of partitions of k into equal parts multiplied by the number of ones in the j-th section of the set of partitions of n, where j = (n - k + 1).

For another version see A245095. - Omar E. Pol, Jul 15 2014

Examples

			For n = 6:
  -------------------------
  k   A000005        T(6,k)
  1      1  *  7   =    7
  2      2  *  5   =   10
  3      2  *  3   =    6
  4      3  *  2   =    6
  5      2  *  1   =    2
  6      4  *  1   =    4
  .         A000041
  -------------------------
So row 6 is [7, 10, 6, 6, 4, 2]. Note that the sum of row 6 is 7+10+6+6+2+4 = 35 equals A006128(6).
.
Triangle begins:
  1;
  1,   2;
  2,   2,  2;
  3,   4,  2,  3;
  5,   6,  4,  3,  2;
  7,  10,  6,  6,  2,  4;
  11, 14, 10,  9,  4,  4,  2;
  15, 22, 14, 15,  6,  8,  2,  4;
  22, 30, 22, 21, 10, 12,  4,  4,  3;
  30, 44, 30, 33, 14, 20,  6,  8,  3,  4;
  42, 60, 44, 45, 22, 28, 10, 12,  6,  4,  2;
  56, 84, 60, 66, 30, 44, 14, 20,  9,  8,  2,  6;
  ...
		

Links

• Paolo Xausa, Table of n, a(n) for n = 1..11325 (rows 1..150 of the triangle, flattened)

Crossrefs

Similar to A221529.

Columns 1-2: A000041, A139582. Leading diagonals 1-3: A000005, A000005, A062011. Row sums give A006128.

Cf. A027293, A135010, A138137, A182703, A245095, A245099.

Programs

Mathematica

A221530row[n_]:=DivisorSigma[0,Range[n]]PartitionsP[n-Range[n]];Array[A221530row,10] (* Paolo Xausa, Sep 04 2023 *)

PARI

row(n) = vector(n, i, numdiv(i)*numbpart(n-i)); \\ Michel Marcus, Jul 18 2014

Formula

T(n,k) = d(k)*p(n-k) = A000005(k)*A027293(n,k).