A323583 Number of ways to split an integer partition of n into consecutive subsequences.

1, 1, 3, 7, 17, 37, 83, 175, 373, 773, 1603, 3275, 6693, 13557, 27447, 55315, 111397, 223769, 449287, 900795, 1805465, 3615929, 7240327, 14491623, 29001625, 58027017, 116093259, 232237583, 464558201, 929224589, 1858623819, 3717475031, 7435314013, 14871103069

Offset

0,3

Links

Table of n, a(n) for n=0..33.

Formula

a(n) = A070933(n)/2.

O.g.f.: (1/2)*Product_{n >= 1} 1/(1 - 2*x^n).

G.f.: 1 + Sum_{k>=1} 2^(k - 1) * x^k / Product_{j=1..k} (1 - x^j). - Ilya Gutkovskiy, Jan 28 2020

Example

The a(3) = 7 ways to split an integer partition of 3 into consecutive subsequences are (3), (21), (2)(1), (111), (11)(1), (1)(11), (1)(1)(1).

Maple

b:= proc(n, i) option remember; `if`(n=0, 1/2, `if`(i<1, 0,
      b(n, i-1) +`if`(i>n, 0, 2*b(n-i, i))))
    end:
a:= n-> ceil(b(n$2)):
seq(a(n), n=0..33);  # Alois P. Heinz, Jan 01 2023

Mathematica

Table[Sum[2^(Length[ptn]-1), {ptn, IntegerPartitions[n]}], {n, 40}]
(* Second program: *)
(1/2) CoefficientList[1 - 1/QPochhammer[2, x] + O[x]^100 , x] (* Jean-François Alcover, Jan 02 2022, after Vladimir Reshetnikov in A070933 *)

Crossrefs

Cf. A006951, A070933, A100883, A279784, A279786, A323433, A323582.

Sequence in context: A111210 A033489 A357212 * A336724 A178941 A178155

Adjacent sequences: A323580 A323581 A323582 * A323584 A323585 A323586

Keyword

nonn

Author

Gus Wiseman, Jan 19 2019

Status

approved