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 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jan 19 2019
STATUS
approved