OFFSET
0,3
COMMENTS
Number of compositions of partitions of n into distinct parts. a(3) = 6: 3, 21, 12, 111, 2|1, 11|1. - Alois P. Heinz, Sep 16 2019
Also the number of ways to split a composition of n into contiguous subsequences with strictly decreasing sums. - Gus Wiseman, Jul 13 2020
This sequence is obtained from the generalized Euler transform in A266964 by taking f(n) = -1, g(n) = (-1) * 2^(n-1). - Seiichi Manyama, Aug 22 2020
LINKS
FORMULA
G.f.: Product_{k>=1} (1 + A011782(k)*x^k).
a(n) ~ 2^n * exp(2*sqrt(-polylog(2, -1/2)*n)) * (-polylog(2, -1/2))^(1/4) / (sqrt(6*Pi) * n^(3/4)). - Vaclav Kotesovec, Sep 19 2019
EXAMPLE
From Gus Wiseman, Jul 13 2020: (Start)
The a(0) = 1 through a(4) = 12 splittings:
() (1) (2) (3) (4)
(1,1) (1,2) (1,3)
(2,1) (2,2)
(1,1,1) (3,1)
(2),(1) (1,1,2)
(1,1),(1) (1,2,1)
(2,1,1)
(3),(1)
(1,1,1,1)
(1,2),(1)
(2,1),(1)
(1,1,1),(1)
(End)
MATHEMATICA
nmax = 33; CoefficientList[Series[Product[(1 + 2^(k - 1) x^k), {k, 1, nmax}], {x, 0, nmax}], x]
PROG
(PARI) N=40; x='x+O('x^N); Vec(prod(k=1, N, 1+2^(k-1)*x^k)) \\ Seiichi Manyama, Aug 22 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, May 22 2018
STATUS
approved