OFFSET
0,3
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)
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..5000
Vaclav Kotesovec, Graph - The asymptotic ratio (100000 terms)
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)
EXAMPLE
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)}.
MATHEMATICA
nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
A072233 counts partitions by sum and length.
KEYWORD
nonn
AUTHOR
Vaclav Kotesovec, Mar 28 2016
STATUS
approved