OFFSET
0,3
COMMENTS
Invert transform of A000009.
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. This interpretation involves only multisets, not sequences. For example, the a(1) = 1 through a(4) = 11 multiset partitions are:
{{1}} {{1,1}} {{1,1,1}} {{1,1,1,1}}
{{1},{2}} {{1},{1,1}} {{1},{1,1,1}}
{{1},{2,2}} {{1,1},{2,2}}
{{2},{1,1}} {{1},{2,2,2}}
{{1},{2},{3}} {{2},{1,1,1}}
{{1},{2},{1,1}}
{{1},{2},{2,2}}
{{1},{2},{3,3}}
{{1},{3},{2,2}}
{{2},{3},{1,1}}
{{1},{2},{3},{4}}
The non-strict version is A055887.
The strongly normal non-strict version is A063834.
The strongly normal version is A270995.
(End)
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..2816
N. J. A. Sloane, Transforms
Eric Weisstein's World of Mathematics, Partition Function Q
FORMULA
G.f.: 1/(1 - Sum_{k>=1} A000009(k)*x^k).
G.f.: 1/(2 - Product_{k>=1} (1 + x^k)).
G.f.: 1/(2 - Product_{k>=1} 1/(1 - x^(2*k-1))).
G.f.: 1/(2 - exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - x^k)))).
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
EXAMPLE
From Gus Wiseman, Jul 31 2022: (Start)
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:
((1)) ((2)) ((3)) ((4))
((1)(1)) ((21)) ((31))
((1)(2)) ((1)(3))
((2)(1)) ((2)(2))
((1)(1)(1)) ((3)(1))
((1)(21))
((21)(1))
((1)(1)(2))
((1)(2)(1))
((2)(1)(1))
((1)(1)(1)(1))
(End)
MAPLE
b:= proc(n) option remember; `if`(n=0, 1, add(b(n-j)*add(
`if`(d::odd, d, 0), d=numtheory[divisors](j)), j=1..n)/n)
end:
a:= proc(n) option remember; `if`(n=0, 1,
add(b(j)*a(n-j), j=1..n))
end:
seq(a(n), n=0..40); # Alois P. Heinz, May 22 2018
MATHEMATICA
nmax = 32; CoefficientList[Series[1/(1 - Sum[PartitionsQ[k] x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - Product[1 + x^k, {k, 1, nmax}]), {x, 0, nmax}], x]
nmax = 32; CoefficientList[Series[1/(2 - 1/QPochhammer[x, x^2]), {x, 0, nmax}], x]
a[0] = 1; a[n_] := a[n] = Sum[PartitionsQ[k] a[n - k], {k, 1, n}]; Table[a[n], {n, 0, 32}]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 22 2018
STATUS
approved