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

%I A270995 #25 Aug 02 2022 22:03:16
%S A270995 1,1,2,4,7,12,23,37,64,108,180,290,488,772,1251,2001,3180,4982,7913,
%T A270995 12261,19162,29669,45804,70187,108029,164276,250267,379439,574067,
%U A270995 864044,1302169,1949050,2917900,4352796,6481627,9620256,14274080,21090608,31142909
%N A270995 Expansion of Product_{k>=1} 1/(1 - A000009(k)*x^k).
%C A270995 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
%C A270995 From _Gus Wiseman_, Jul 31 2022: (Start)
%C A270995 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:
%C A270995   {{1}}  {{1,1}}    {{1,1,1}}      {{1,1,1,1}}
%C A270995          {{1},{2}}  {{1},{1,1}}    {{1},{1,1,1}}
%C A270995                     {{2},{1,1}}    {{1,1},{2,2}}
%C A270995                     {{1},{2},{3}}  {{2},{1,1,1}}
%C A270995                                    {{1},{2},{1,1}}
%C A270995                                    {{2},{3},{1,1}}
%C A270995                                    {{1},{2},{3},{4}}
%C A270995 The weakly normal non-strict version is A055887.
%C A270995 The non-strict version is A063834.
%C A270995 The weakly normal version is A304969.
%C A270995 (End)
%H A270995 Vaclav Kotesovec, <a href="/A270995/b270995.txt">Table of n, a(n) for n = 0..5000</a>
%H A270995 Vaclav Kotesovec, <a href="/A270995/a270995.jpg">Graph - The asymptotic ratio (100000 terms)</a>
%F A270995 From _Vaclav Kotesovec_, Mar 28 2016: (Start)
%F A270995 a(n) ~ c * n^2 * 2^(n/3), where
%F A270995 c = 436246966131366188.9451742926272200575837456478739... if mod(n,3) = 0
%F A270995 c = 436246966131366188.9351143199611598469443841182807... if mod(n,3) = 1
%F A270995 c = 436246966131366188.9322714926383227135786894927498... if mod(n,3) = 2
%F A270995 (End)
%e A270995 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)}.
%t A270995 nmax = 50; CoefficientList[Series[Product[1/(1-PartitionsQ[k]*x^k), {k, 1, nmax}], {x, 0, nmax}], x]
%Y A270995 Cf. A063834 (twice partitioned numbers), A271619, A279784, A327554, A327608.
%Y A270995 The unordered version is A089259, non-strict A001970 (row-sums of A061260).
%Y A270995 For compositions instead of partitions we have A304969, non-strict A055887.
%Y A270995 A000041 counts integer partitions, strict A000009.
%Y A270995 A072233 counts partitions by sum and length.
%K A270995 nonn
%O A270995 0,3
%A A270995 _Vaclav Kotesovec_, Mar 28 2016