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.

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

1,2

Comments

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

Links

Table of n, a(n) for n=1..95.

Formula

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

Example

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.

Crossrefs

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.

Sequence in context: A019170 A107504 A141674 * A178394 A266391 A091664

Adjacent sequences: A329320 A329321 A329322 * A329324 A329325 A329326

Keyword

nonn,tabl

Author

Omar E. Pol, Nov 21 2019

Status

approved