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Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).
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%I #27 Jan 25 2024 08:02:53

%S 1,1,2,6,14,34,88,216,532,1322,3290,8142,20192,50080,124144,307878,

%T 763474,1893038,4694060,11639580,28861736,71567206,177460750,

%U 440037738,1091134276,2705618900,6708953156,16635775698,41250705518,102286806130,253634237896,628921097352,1559496588628

%N Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).

%C Invert transform of A032020.

%C Number of ways to choose a strict composition of each part of a composition of n. - _Gus Wiseman_, Jul 18 2020

%C 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

%H Alois P. Heinz, <a href="/A307068/b307068.txt">Table of n, a(n) for n = 0..2536</a>

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

%e From _Gus Wiseman_, Jul 18 2020: (Start)

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

%e (1) (2) (3) (4)

%e (1),(1) (1,2) (1,3)

%e (2,1) (3,1)

%e (1),(2) (1),(3)

%e (2),(1) (2),(2)

%e (1),(1),(1) (3),(1)

%e (1),(1,2)

%e (1),(2,1)

%e (1,2),(1)

%e (2,1),(1)

%e (1),(1),(2)

%e (1),(2),(1)

%e (2),(1),(1)

%e (1),(1),(1),(1)

%e (End)

%p T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,

%p `if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))

%p end:

%p g:= proc(n) option remember; add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)) end:

%p a:= proc(n) option remember; `if`(n<1, 1,

%p add(a(n-i)*g(i), i=1..n))

%p end:

%p seq(a(n), n=0..32); # _Alois P. Heinz_, Dec 16 2022

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

%o (Magma)

%o m:=80;

%o R<x>:=PowerSeriesRing(Integers(), m);

%o Coefficients(R!( 1/(1 - (&+[Factorial(k)*x^Binomial(k+1,2)/(&*[ 1-x^j: j in [1..k]]): k in [1..m+2]]) ) )); // _G. C. Greubel_, Jan 25 2024

%o (SageMath)

%o m=80;

%o def p(x, j): return product(1-x^k for k in range(1,j+1))

%o def f(x): return 1/(1 - sum(factorial(j)*x^binomial(j+1,2)/p(x,j) for j in range(1, m+3)) )

%o def A307068_list(prec):

%o P.<x> = PowerSeriesRing(QQ, prec)

%o return P( f(x) ).list()

%o A307068_list(m) # _G. C. Greubel_, Jan 25 2024

%Y The version for partitions is A270995.

%Y Starting with a strict composition gives A336139.

%Y Strict compositions are counted by A032020.

%Y Partitions of each part of a partition are A063834.

%Y Compositions of each part of a partition are A075900.

%Y Compositions of each part of a composition are A133494.

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

%Y Compositions of each part of a strict partition are A304961.

%Y Strict partitions of each part of a composition are A304969.

%Y Compositions of each part of a strict composition are A336127.

%Y Set partitions of strict compositions are A336140.

%Y Strict compositions of each part of a partition are A336141.

%Y Cf. A001970, A307067, A317536, A318683, A318684, A319794, A323583, A336128, A336130, A336132.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Mar 22 2019