A306489 Square array A(n,k), n >= 0, k >= 1, read by antidiagonals, where column k is the expansion of 1/(1 - Sum_{d|k} x^d).

1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 1, 1, 1, 2, 2, 5, 1, 1, 1, 1, 3, 3, 8, 1, 1, 1, 2, 1, 6, 4, 13, 1, 1, 1, 1, 4, 1, 10, 6, 21, 1, 1, 1, 2, 1, 7, 2, 18, 9, 34, 1, 1, 1, 1, 3, 1, 13, 3, 31, 13, 55, 1, 1, 1, 2, 2, 6, 1, 25, 4, 55, 19, 89, 1, 1, 1, 1, 3, 3, 10, 1, 46, 5, 96, 28, 144, 1

Offset

0,9

Comments

A(n,k) is the number of compositions (ordered partitions) of n into divisors of k.

Links

Table of n, a(n) for n=0..90.

Formula

G.f. of column k: 1/(1 - Sum_{d|k} x^d).

Example

Square array begins:
  1,  1,  1,   1,  1,   1,  ...
  1,  1,  1,   1,  1,   1,  ...
  1,  2,  1,   2,  1,   2,  ...
  1,  3,  2,   3,  1,   4,  ...
  1,  5,  3,   6,  1,   7,  ...
  1,  8,  4,  10,  2,  13,  ...

Mathematica

Table[Function[k, SeriesCoefficient[1/(1 - Sum[x^d, {d, Divisors[k]}]), {x, 0, n}]][i - n + 1], {i, 0, 12}, {n, 0, i}] // Flatten

Crossrefs

Columns k=1..7 give A000012, A000045 (for n > 0), A000930, A060945, A003520, A079958, A005709.

Cf. A100346, A214575.

Sequence in context: A371212 A250261 A063669 * A319734 A211005 A162154

Adjacent sequences: A306486 A306487 A306488 * A306490 A306491 A306492

Keyword

nonn,tabl

Author

Ilya Gutkovskiy, Feb 19 2019

Status

approved