A327096 -id:A327096 - cp's OEIS frontend

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

1, 3, 1, 4, 0, 1, 7, 3, 0, 1, 6, 0, 0, 0, 1, 12, 4, 3, 0, 0, 1, 8, 0, 0, 0, 0, 0, 1, 15, 7, 0, 3, 0, 0, 0, 1, 13, 0, 4, 0, 0, 0, 0, 0, 1, 18, 6, 0, 0, 3, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1, 14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 24, 8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1

Offset: 1

Gary W. Adamson, Jun 03 2007

The original definition was: A127093 * A125093^(-1).
Left border = A000203, sigma(n): (1, 3, 4, 7, 6, ...). Row sums = A007429: (1, 4, 5, 11, 7, 20, 9, ...); = inverse Moebius transform applied to sigma(n); (i.e., inverse Moebius transform applied twice to natural numbers).
T(n,k) is the total number of parts congruent to 0 mod k in the partitions of n into equal parts. - Omar E. Pol, Nov 19 2019
From Omar E. Pol, Jan 01 2020: (Start)
Conjecture 1: the sum of odd-indexed terms in row n equals A327096(n).
Conjecture 2: the sum of even-indexed terms in row n equals the n-th term of the sequence formed by A000004 and A007429 interleaved.
Conjecture 3: alternating row sums give A288417. (End)

			First few rows of the triangle are:
   1;
   3,  1;
   4,  0, 1;
   7,  3, 0, 1;
   6,  0, 0, 0, 1;
  12,  4, 3, 0, 0, 1;
   8,  0, 0, 0, 0, 0, 1;
  15,  7, 0, 3, 0, 0, 0, 1;
  13,  0, 4, 0, 0, 0, 0, 0, 1;
  18,  6, 0, 0, 3, 0, 0, 0, 0, 1;
  12,  0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  28, 12, 7, 4, 0, 3, 0, 0, 0, 0, 0, 1;
  14,  0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  24,  8, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 1;
  24,  0, 6, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
  31, 15, 0, 7, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 0, 1;
...
Extended by _Omar E. Pol_, Nov 19 2019

Cf. A000203, A007429, A127093, A125093, A244051, A288417, A327096.

A127093 * A125093^(-1), as infinite lower triangular matrices.

New name and more terms from Omar E. Pol, Nov 19 2019

cp's OEIS Frontend

A130540 Triangle read by rows T(n,k) in which column k lists the terms of A000203 interspersed with (k-1) zeros, 1 <= k <= n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Formula

Extensions