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A307068 Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)). 6
1, 1, 2, 6, 14, 34, 88, 216, 532, 1322, 3290, 8142, 20192, 50080, 124144, 307878, 763474, 1893038, 4694060, 11639580, 28861736, 71567206, 177460750, 440037738, 1091134276, 2705618900, 6708953156, 16635775698, 41250705518, 102286806130, 253634237896, 628921097352, 1559496588628 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

Invert transform of A032020.

Number of ways to choose a strict composition of each part of a composition of n. - Gus Wiseman, Jul 18 2020

LINKS

Table of n, a(n) for n=0..32.

FORMULA

a(0) = 1; a(n) = Sum_{k=1..n} A032020(k)*a(n-k).

The Invert transform T(a) of a sequence a is given by T(a)_n = Sum_c Product_i a(c_i), where the sum is over all compositions c of n. - Gus Wiseman, Aug 01 2020

EXAMPLE

From Gus Wiseman, Jul 18 2020: (Start)

The a(1) = 1 through a(4) = 14 ways to choose a strict composition of each part of a composition:

    (1)  (2)      (3)          (4)

         (1),(1)  (1,2)        (1,3)

                  (2,1)        (3,1)

                  (1),(2)      (1),(3)

                  (2),(1)      (2),(2)

                  (1),(1),(1)  (3),(1)

                               (1),(1,2)

                               (1),(2,1)

                               (1,2),(1)

                               (2,1),(1)

                               (1),(1),(2)

                               (1),(2),(1)

                               (2),(1),(1)

                               (1),(1),(1),(1)

(End)

MATHEMATICA

nmax = 32; CoefficientList[Series[1/(1 - Sum[k! x^(k (k + 1)/2)/Product[(1 - x^j), {j, 1, k}], {k, 1, nmax}]), {x, 0, nmax}], x]

CROSSREFS

Cf. A307067, A317536.

The version for partitions is A270995.

Starting with a strict composition gives A336139.

Strict compositions are counted by A032020.

Partitions of each part of a partition are A063834.

Compositions of each part of a partition are A075900.

Compositions of each part of a composition are A133494.

Strict partitions of each part of a strict partition are A279785.

Compositions of each part of a strict partition are A304961.

Strict partitions of each part of a composition are A304969.

Compositions of each part of a strict composition are A336127.

Set partitions of strict compositions are A336140.

Strict compositions of each part of a partition are A336141.

Cf. A001970, A318683, A318684, A319794, A323583, A336128, A336130, A336132.

Sequence in context: A099425 A186523 A177790 * A269506 A292816 A105635

Adjacent sequences:  A307065 A307066 A307067 * A307069 A307070 A307071

KEYWORD

nonn

AUTHOR

Ilya Gutkovskiy, Mar 22 2019

STATUS

approved

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Last modified March 4 05:42 EST 2021. Contains 341779 sequences. (Running on oeis4.)