A307068 Expansion of 1/(1 - Sum_{k>=1} k!*x^(k*(k+1)/2) / Product_{j=1..k} (1 - x^j)).

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

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

• nonn

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

Examples

			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)
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..2536

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.

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

Programs

Magma

m:=80;
R:=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

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]

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. = PowerSeriesRing(QQ, prec)
    return P( f(x) ).list()
A307068_list(m) # G. C. Greubel, Jan 25 2024

Formula

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