|
%I #20 Aug 09 2022 23:16:59
%S 1,1,2,5,11,25,57,129,292,662,1500,3398,7699,17443,39519,89536,202855,
%T 459593,1041267,2359122,5344889,12109524,27435660,62158961,140828999,
%U 319065932,722884274,1637785870,3710611298,8406859805,19046805534,43152950024,97768473163
%N Expansion of 1/(1 - Sum_{k>=1} q(k)*x^k), where q(k) = number of partitions of k into distinct parts (A000009).
%C Invert transform of A000009.
%C From _Gus Wiseman_, Jul 31 2022: (Start)
%C 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. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
%C {{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
%C {{1},{2}} {{1},{1,1}} {{1},{1,1,1}}
%C {{1},{2,2}} {{1,1},{2,2}}
%C {{2},{1,1}} {{1},{2,2,2}}
%C {{1},{2},{3}} {{2},{1,1,1}}
%C {{1},{2},{1,1}}
%C {{1},{2},{2,2}}
%C {{1},{2},{3,3}}
%C {{1},{3},{2,2}}
%C {{2},{3},{1,1}}
%C {{1},{2},{3},{4}}
%C The non-strict version is A055887.
%C The strongly normal non-strict version is A063834.
%C The strongly normal version is A270995.
%C (End)
%H Alois P. Heinz, <a href="/A304969/b304969.txt">Table of n, a(n) for n = 0..2816</a>
%H N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PartitionFunctionQ.html">Partition Function Q</a>
%H <a href="/index/Par#part">Index entries for sequences related to partitions</a>
%H <a href="/index/Com#comp">Index entries for sequences related to compositions</a>
%F G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
%F G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
%F G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
%F G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
%F a(n) ~ c / r^n, where r = 0.441378990861652015438479635503868737167721352874... is the root of the equation QPochhammer[-1, r] = 4 and c = 0.4208931614610039677452560636348863586180784719323982664940444607322... - _Vaclav Kotesovec_, May 23 2018
%e From _Gus Wiseman_, Jul 31 2022: (Start)
%e a(n) is the number of ways to choose a strict integer partition of each part of an integer composition of n. The a(1) = 1 through a(4) = 11 choices are:
%e ((1)) ((2)) ((3)) ((4))
%e ((1)(1)) ((21)) ((31))
%e ((1)(2)) ((1)(3))
%e ((2)(1)) ((2)(2))
%e ((1)(1)(1)) ((3)(1))
%e ((1)(21))
%e ((21)(1))
%e ((1)(1)(2))
%e ((1)(2)(1))
%e ((2)(1)(1))
%e ((1)(1)(1)(1))
%e (End)
%p b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
%p `if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
%p end:
%p a:= proc(n) option remember; `if`(n=0, 1,
%p add(b(j)*a(n-j), j=1..n))
%p end:
%p seq(a(n), n=0..40); # _Alois P. Heinz_, May 22 2018
%t nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
%t nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
%t a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]
%Y Cf. A050342, A055887, A067687, A081362, A279785, A299106.
%Y Row sums of A308680.
%Y The unordered version is A089259, non-strict A001970 (row-sums of A061260).
%Y For partitions instead of compositions we have A270995, non-strict A063834.
%Y A000041 counts integer partitions, strict A000009.
%Y A072233 counts partitions by sum and length.
%Y Cf. A279784.
%K nonn
%O 0,3
%A _Ilya Gutkovskiy_, May 22 2018
|
