A130540 -id:A130540 - cp's OEIS frontend

A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

1, 4, 1, 5, 1, 1, 11, 4, 1, 1, 7, 1, 1, 1, 1, 20, 8, 4, 1, 1, 19, 1, 1, 1, 1, 1, 1, 26, 11, 4, 4, 1, 1, 1, 1, 18, 5, 5, 1, 1, 1, 1, 1, 1, 28, 10, 4, 4, 4, 1, 1, 1, 1, 1, 13, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 55, 27, 15, 8, 4, 4, 1, 1, 1, 1, 1, 1, 15, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1

Offset: 1

Gary W. Adamson, Aug 06 2008

Left border = A007429: (1, 4, 5, 11, 7, 20, 9,...).

Row sums = A060640: (1, 5, 7, 17, 11, 35,...).

			First few rows of the triangle =
1;
4, 1;
5, 1, 1;
11, 4, 1, 1;
7, 1, 1, 1, 1;
20, 8, 4, 1, 1, 1;
9, 1, 1, 1, 1, 1, 1;
...
Row 4 = (11, 4, 1, 1) since row 4 of A130540 = (7, 3, 0, 1).

Cf. A130540, A007429, A060640.

Triangle read by rows, A130540 * A000012, 1<=k<=n. Equals partial row sums of A130540 starting from the right.

A329323 Triangle read by rows: T(n,k) is the sum of the parts congruent to 0 mod k in the partitions of n into equal parts, 1 <= k <= n.

Original entry on oeis.org

1, 4, 2, 6, 0, 3, 12, 8, 0, 4, 10, 0, 0, 0, 5, 24, 12, 12, 0, 0, 6, 14, 0, 0, 0, 0, 0, 7, 32, 24, 0, 16, 0, 0, 0, 8, 27, 0, 18, 0, 0, 0, 0, 0, 9, 40, 20, 0, 0, 20, 0, 0, 0, 0, 10, 22, 0, 0, 0, 0, 0, 0, 0, 0, 0, 11, 72, 48, 36, 24, 0, 24, 0, 0, 0, 0, 0, 12, 26, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 56, 28, 0, 0

Offset: 1

Omar E. Pol, Nov 21 2019

Column k lists the terms of A038040 multiplied by k and interspersed with (k-1) zeros.

			Triangle begins:
   1;
   4,  2;
   6,  0,  3;
  12,  8,  0,  4;
  10,  0,  0,  0,  5;
  24, 12, 12,  0,  0,  6;
  14,  0,  0,  0,  0,  0,  7;
  32, 24,  0, 16,  0,  0,  0,  8;
  27,  0, 18,  0,  0,  0,  0,  0,  9;
  40, 20,  0,  0, 20,  0,  0,  0,  0, 10;
  22,  0,  0,  0,  0,  0,  0,  0,  0,  0, 11;
  72, 48, 36, 24,  0, 24,  0,  0,  0,  0,  0, 12;
  26,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0,  0, 13;
  56, 28,  0,  0,  0,  0, 28,  0,  0,  0,  0,  0,  0, 14;
  60,  0, 30,  0, 30,  0,  0,  0,  0,  0,  0,  0,  0,  0, 15;
  80, 64,  0, 48,  0,  0,  0, 32,  0,  0,  0,  0,  0,  0,  0, 16;
...
For n = 6 the partitions of 6 into equal parts are [6], [3, 3], [2, 2, 2], [1, 1, 1, 1, 1, 1]. Then, for k = 2 the sum of the parts that are multiples of 2 is 6 + 2 + 2 + 2 = 12, so T(6,2) = 12.

Column 1 is A038040.
Row sums give A034718.
Leading diagonal gives A000027.
The number of positive terms in row n is A000005(n).
Cf. A126988, A130540, A134577, A244051.

T(n,k) = A126988(n,k)*A134577(n,k).

cp's OEIS Frontend

A143313 Triangle read by rows, A130540 * A000012, 1<=k<=n.

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