A098133 -id:A098133 - cp's OEIS frontend

A339446 Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.

1, 0, 0, 0, 2, 2, 2, 2, 2, 2, 2, 8, 8, 14, 14, 20, 20, 26, 26, 32, 32, 62, 62, 92, 116, 146, 170, 224, 248, 302, 350, 404, 452, 530, 698, 776, 968, 1166, 1478, 1700, 2132, 2474, 3050, 3512, 4208, 4814, 5750, 6476, 7556, 8522, 10562, 11672, 13952, 16022, 19286, 22316, 26540

Offset: 1

Ilya Gutkovskiy, Dec 05 2020

nonn

			a(12) = 8 because we have [10, 2], [2, 10], [5, 4, 3], [5, 3, 4], [4, 5, 3], [4, 3, 5], [3, 5, 4] and [3, 4, 5].

Seiichi Manyama, Table of n, a(n) for n = 1..1000
Index entries for sequences related to compositions

Cf. A006141, A032020, A096401, A098133.

nmax = 60; CoefficientList[Series[Sum[k! x^(k (3 k - 1)/2)/Product[1 - x^j, {j, 1, k - 1}], {k, 1, nmax}], {x, 0, nmax}], x] // Rest

G.f.: Sum_{k>=1} k! * x^(k*(3*k - 1)/2) / Product_{j=1..k-1} (1 - x^j).

A348124 Number of compositions of n where the smallest part is smaller than the number of parts.

Original entry on oeis.org

0, 1, 3, 6, 13, 28, 59, 122, 248, 501, 1009, 2028, 4070, 8159, 16343, 32717, 65472, 130991, 262041, 524157, 1048410, 2096943, 4194043, 8388285, 16776819, 33553946, 67108270, 134217002, 268434568, 536869825, 1073740493, 2147482019, 4294965305, 8589932164

Offset: 1

R. J. Mathar, Oct 01 2021

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A011782, A098132, A098133.

b:= proc(n, s, c) option remember; `if`(s b(n$2, 0):
seq(a(n), n=1..40);  # Alois P. Heinz, Oct 01 2021

b[n_, s_, c_] := b[n, s, c] = If[s < c, Ceiling[2^(n - 1)],
     If[n == 0, 0, Sum[b[n - j, Min[j, s], c + 1], {j, 1, n}]]];
a[n_] := b[n, n, 0];
Table[a[n], {n, 1, 40}] (* Jean-François Alcover, Apr 14 2022, after Alois P. Heinz *)

a(n) + A098132(n) + A098133(n) = 2^(n-1).

a(23)-a(34) from Alois P. Heinz, Oct 01 2021

cp's OEIS Frontend

A339446 Number of compositions (ordered partitions) of n into distinct parts such that the smallest part is equal to the number of parts.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Formula