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This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

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A336139 Number of ways to choose a strict composition of each part of a strict composition of n.

Original entry on oeis.org

1, 1, 1, 5, 9, 17, 45, 81, 181, 397, 965, 1729, 3673, 7313, 15401, 34065, 68617, 135069, 266701, 556969, 1061921, 2434385, 4436157, 9120869, 17811665, 35651301, 68949549, 136796317, 283612973, 537616261, 1039994921, 2081261717, 3980842425, 7723253181, 15027216049
Offset: 0

Views

Author

Gus Wiseman, Jul 16 2020

Keywords

Comments

A strict composition of n is a finite sequence of distinct positive integers summing to n.

Examples

			The a(1) = 1 through a(5) = 17 splittings:
  (1)  (2)  (3)      (4)        (5)
            (1,2)    (1,3)      (1,4)
            (2,1)    (3,1)      (2,3)
            (1),(2)  (1),(3)    (3,2)
            (2),(1)  (3),(1)    (4,1)
                     (1),(1,2)  (1),(4)
                     (1),(2,1)  (2),(3)
                     (1,2),(1)  (3),(2)
                     (2,1),(1)  (4),(1)
                                (1),(1,3)
                                (1,2),(2)
                                (1),(3,1)
                                (1,3),(1)
                                (2),(1,2)
                                (2,1),(2)
                                (2),(2,1)
                                (3,1),(1)

Links

Crossrefs

The version for partitions is A063834.
Row sums of A072574.
The version for non-strict compositions is A133494.
The version for strict partitions is A279785.
Multiset partitions of partitions are A001970.
Strict compositions are A032020.
Taking a composition of each part of a partition: A075900.
Taking a composition of each part of a strict partition: A304961.
Taking a strict composition of each part of a composition: A307068.
Splittings of partitions are A323583.
Compositions of parts of strict compositions are A336127.
Set partitions of strict compositions are A336140.
Cf. A318683, A318684, A319794, A336128, A336130, A336132.

Programs

  • Mathematica
    strs[n_]:=Join@@Permutations/@Select[IntegerPartitions[n],UnsameQ@@#&];
Table[Length[Join@@Table[Tuples[strs/@ctn],{ctn,strs[n]}]],{n,0,15}]

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

Original entry on oeis.org

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

Views

Author

Ilya Gutkovskiy, Mar 22 2019

Keywords

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

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).
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