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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. 2
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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)

LINKS

Table of n, a(n) for n=1..105.

FORMULA

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

EXAMPLE

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

CROSSREFS

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

Sequence in context: A174382 A123730 A143317 * A076816 A021765 A267187

Adjacent sequences:  A130537 A130538 A130539 * A130541 A130542 A130543

KEYWORD

nonn,tabl

AUTHOR

Gary W. Adamson, Jun 03 2007

EXTENSIONS

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

STATUS

approved

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Last modified May 11 13:41 EDT 2021. Contains 343791 sequences. (Running on oeis4.)