A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).

1, 1, 2, 4, 7, 12, 23, 37, 64, 108, 180, 290, 488, 772, 1251, 2001, 3180, 4982, 7913, 12261, 19162, 29669, 45804, 70187, 108029, 164276, 250267, 379439, 574067, 864044, 1302169, 1949050, 2917900, 4352796, 6481627, 9620256, 14274080, 21090608, 31142909

Offset: 0

Author

Vaclav Kotesovec, Mar 28 2016

Keywords

• nonn

Comments

The number of ways a number can be partitioned into not necessarily distinct parts and then each part is partitioned into distinct parts. Also a(n) > A089259(n) for n>5. - Gus Wiseman, Apr 10 2016

From Gus Wiseman, Jul 31 2022: (Start)

Also the number of ways to choose a multiset partition into distinct constant multisets of a multiset of length n that covers an initial interval of positive integers with weakly decreasing multiplicities. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 7 multiset partitions are:

{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}

{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}

{{2},{1,1}} {{1,1},{2,2}}

{{1},{2},{3}} {{2},{1,1,1}}

{{1},{2},{1,1}}

{{2},{3},{1,1}}

{{1},{2},{3},{4}}

The weakly normal non-strict version is A055887.

The non-strict version is A063834.

The weakly normal version is A304969.

(End)

Examples

			a(6)=23: {(6), (5)(1), (51), (4)(2), (42), (4)(1)(1), (41)(1), (3)(3), (3)(2)(1), (3)(21), (32)(1), (31)(2), (21)(3), (321), (3)(1)(1)(1), (31)(1)(1), (2)(2)(2), (2)(2)(1)(1), (21)(2)(1), (21)(21), (2)(1)(1)(1)(1), (21)(1)(1)(1), (1)(1)(1)(1)(1)(1)}.
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
• Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)

Crossrefs

Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.

The unordered version is A089259, non-strict A001970 (row-sums of A061260).

For compositions instead of partitions we have A304969, non-strict A055887.

A000041 counts integer partitions, strict A000009.

A072233 counts partitions by sum and length.

Programs

Mathematica

nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

From Vaclav Kotesovec, Mar 28 2016: (Start)

a(n) ~ c * n^2 * 2^(n/3), where

c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0

c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1

c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2

(End)