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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)).
10
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
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
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
LINKS
FORMULA
a(0) = 1; a(n) = Sum_{k=1..n} A032020(k)*a(n-k).
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)
MAPLE
T:= proc(n, k) option remember; `if`(k<0 or n<0, 0,
`if`(k=0, `if`(n=0, 1, 0), T(n-k, k) +k*T(n-k, k-1)))
end:
g:= proc(n) option remember; add(T(n, k), k=0..floor((sqrt(8*n+1)-1)/2)) end:
a:= proc(n) option remember; `if`(n<1, 1,
add(a(n-i)*g(i), i=1..n))
end:
seq(a(n), n=0..32); # Alois P. Heinz, Dec 16 2022
MATHEMATICA
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]
PROG
(Magma)
m:=80;
R<x>:=PowerSeriesRing(Integers(), m);
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
(SageMath)
m=80;
def p(x, j): return product(1-x^k for k in range(1, j+1))
def f(x): return 1/(1 - sum(factorial(j)*x^binomial(j+1, 2)/p(x, j) for j in range(1, m+3)) )
def A307068_list(prec):
P.<x> = PowerSeriesRing(QQ, prec)
return P( f(x) ).list()
A307068_list(m) # G. C. Greubel, Jan 25 2024
CROSSREFS
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.
Sequence in context: A099425 A186523 A177790 * A269506 A292816 A105635
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Mar 22 2019
STATUS
approved