A334467 Square array read by antidiagonals upwards: T(n,k) is the sum of all parts of all partitions of n into consecutive parts that differ by k, with n >= 1, k >= 0.

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

Offset

1,2

Links

Table of n, a(n) for n=1..91.

Formula

T(n,k) = n*A323345(n,k).

Example

Array begins:
     k  0   1   2   3   4   5   6   7   8   9  10
   n +------------------------------------------------
   1 |  1,  1,  1,  1,  1,  1,  1,  1,  1,  1,  1, ...
   2 |  4,  2,  2,  2,  2,  2,  2,  2,  2,  2,  2, ...
   3 |  6,  6,  3,  3,  3,  3,  3,  3,  3,  3,  3, ...
   4 | 12,  4,  8,  4,  4,  4,  4,  4,  4,  4,  4, ...
   5 | 10, 10,  5, 10,  5,  5,  5,  5,  5,  5,  5, ...
   6 | 24, 12, 12,  6, 12,  6,  6,  6,  6,  6,  6, ...
   7 | 14, 14,  7, 14,  7, 14,  7,  7,  7,  7,  7, ...
   8 | 32,  8, 16,  8, 16,  8, 16,  8,  8,  8,  8, ...
   9 | 27, 27, 18, 18,  9, 18,  9, 18,  9,  9,  9, ...
  10 | 40, 20, 20, 10, 20, 20, 20, 10, 20, 10, 10, ...
...

Mathematica

nmax = 13;
col[k_] := col[k] = CoefficientList[Sum[x^(n(k n - k + 2)/2 - 1)/(1 - x^n), {n, 1, nmax}] + O[x]^nmax, x];
T[n_, k_] := n col[k][[n]];
Table[T[n-k, k], {n, 1, nmax}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Nov 30 2020 *)

Crossrefs

Columns k: A038040 (k=0), A245579 (k=1), A060872 (k=2), A334463 (k=3), A327262 (k=4), A334733 (k=5), A334953 (k=6).

Every diagonal starting with 1 gives A000027.

Sequences of number of parts related to column k: A000203 (k=0), A204217 (k=1), A066839 (k=2) (conjectured), A330889 (k=3), A334464 (k=4), A334732 (k=5), A334949 (k=6).

Sequences of number of partitions related to column k: A000005 (k=0), A001227 (k=1), A038548 (k=2), A117277 (k=3), A334461 (k=4), A334541 (k=5), A334948 (k=6).

Polygonal numbers related to column k: A001477 (k=0), A000217 (k=1), A000290 (k=2), A000326 (k=3), A000384 (k=4), A000566 (k=5), A000567 (k=6).

Cf. A245579, A323345, A334466.

Sequence in context: A032145 A032050 A109915 * A255198 A202521 A247362

Adjacent sequences: A334464 A334465 A334466 * A334468 A334469 A334470

Keyword

nonn,tabl

Author

Omar E. Pol, May 05 2020

Status

approved