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

Author

Gus Wiseman, Jan 19 2019

Keywords

• nonn

Examples

			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).
		

Crossrefs

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

Programs

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 *)

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