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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
OFFSET
0,3
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 *)
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 19 2019
STATUS
approved