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A332304 Number of compositions (ordered partitions) of n into distinct parts such that number of parts is odd. 14
0, 1, 1, 1, 1, 1, 7, 7, 13, 19, 25, 31, 43, 49, 61, 193, 205, 337, 475, 727, 985, 1363, 1741, 2359, 2983, 3841, 4705, 5929, 12193, 13777, 20527, 27631, 39901, 52651, 75601, 99151, 132907, 172297, 227053, 287569, 373525, 465241, 587563, 725839, 899761, 1457683 (list; graph; refs; listen; history; text; internal format)
OFFSET
0,7
LINKS
FORMULA
G.f.: Sum_{k>=1} (2*k - 1)! * x^(k*(2*k - 1)) / Product_{j=1..2*k-1} (1 - x^j).
a(n) = A032020(n) - A332305(n).
EXAMPLE
a(6) = 7 because we have [6], [3, 2, 1], [3, 1, 2], [2, 3, 1], [2, 1, 3], [1, 3, 2] and [1, 2, 3].
MAPLE
b:= proc(n, i, p) option remember; `if`(i*(i+1)/2<n, 0, `if`(n=0,
irem(p, 2)*p!, add(b(n-i*j, i-1, p+j), j=0..min(1, n/i))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55); # Alois P. Heinz, Feb 09 2020
MATHEMATICA
nmax = 45; CoefficientList[Series[Sum[(2 k - 1)! x^(k (2 k - 1))/Product[1 - x^j, {j, 1, 2 k - 1}], {k, 1, nmax}], {x, 0, nmax}], x]
CROSSREFS
Sequence in context: A335895 A072821 A038589 * A317790 A109539 A109541
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Feb 09 2020
STATUS
approved

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Last modified March 29 00:26 EDT 2024. Contains 371264 sequences. (Running on oeis4.)