A304961 Expansion of Product_{k>=1} (1 + 2^(k-1)*x^k).

1, 1, 2, 6, 12, 32, 72, 176, 384, 960, 2112, 4992, 11264, 26112, 58368, 136192, 301056, 688128, 1548288, 3489792, 7766016, 17596416, 38993920, 87293952, 194248704, 432537600, 957349888, 2132803584, 4699717632, 10406068224, 23001563136, 50683969536, 111434268672, 245819768832

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

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Index entries for sequences related to partitions

Index entries for sequences related to compositions

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

Cf. A000009, A011782, A022629, A098407, A102866, A266964, A271619, A279785.

The non-strict version is A075900.

Starting with a reversed partition gives A323583.

Starting with a partition gives A336134.

Partitions of partitions are A001970.

Splittings with equal sums are A074854.

Splittings of compositions are A133494.

Splittings with distinct sums are A336127.

Cf. A006951, A063834, A316245, A317715, A319794, A323582, A336135, A336136.

Sequence in context: A163087 A332654 A000650 * A032178 A102881 A360867

Adjacent sequences: A304958 A304959 A304960 * A304962 A304963 A304964

Keyword

nonn

Author

Ilya Gutkovskiy, May 22 2018

Status

approved