A303939 -id:A303939 - cp's OEIS frontend

A304036 Number of partitions of n into at most 2 copies of 1!, 3 copies of 2!, 4 copies of 3!, ... .

1, 1, 2, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2, 1, 3, 2, 3, 1, 2, 1, 4, 3, 5, 2, 4, 2, 5, 3, 4, 1, 2, 1, 3, 2, 3, 1, 2

Offset: 0

Seiichi Manyama, May 05 2018

			a(6) = 3 because we have [6], [2,2,2] and [2,2,1,1].

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Wikipedia, Factorial number system

Cf. A001048, A064986, A303939, A304039.

G.f.: Product_{j>=1} Sum_{k=0..j+1} x^(k*j!) = Product_{j>=1} (1-x^((j+1)!+j!))/(1-x^(j!)).

A303940 Square array A(n,k), n >= 0, k >= 0, read by antidiagonals, where column k is the expansion of Product_{j>=1} (1-x^(j*(j+k)))/(1-x^j). in powers of x.

Original entry on oeis.org

1, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 2, 2, 3, 2, 1, 1, 2, 3, 4, 4, 3, 1, 1, 2, 3, 4, 5, 5, 3, 1, 1, 2, 3, 5, 6, 8, 7, 5, 1, 1, 2, 3, 5, 6, 9, 10, 10, 5, 1, 1, 2, 3, 5, 7, 10, 12, 14, 13, 8, 1, 1, 2, 3, 5, 7, 10, 13, 17, 18, 17, 9, 1, 1, 2, 3, 5, 7, 11, 14, 19, 23, 25, 22, 13

Offset: 0

Seiichi Manyama, May 03 2018

A(n,k) is the number of partitions of n into at most 0+k copies of 1, 1+k copies of 2, 2+k copies of 3, ... .

			Square array begins:
   1, 1, 1, 1,  1,  1,  1,  1, ...
   0, 1, 1, 1,  1,  1,  1,  1, ...
   1, 1, 2, 2,  2,  2,  2,  2, ...
   1, 2, 2, 3,  3,  3,  3,  3, ...
   1, 3, 4, 4,  5,  5,  5,  5, ...
   2, 4, 5, 6,  6,  7,  7,  7, ...
   3, 5, 8, 9, 10, 10, 11, 11, ...

Seiichi Manyama, Antidiagonals n = 0..139, flattened

Columns k=0..2 give A087153, A052335, A303939.

Main diagonal gives A000041.

cp's OEIS Frontend

A304036 Number of partitions of n into at most 2 copies of 1!, 3 copies of 2!, 4 copies of 3!, ... .

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