|
| |
|
|
A323583
|
|
Number of ways to split an integer partition of n into consecutive subsequences.
|
|
26
|
|
|
|
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
(list;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
|
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
|
| |
|
|
