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A339407
Number of partitions of n into an odd number of parts that are not multiples of 4.
3
0, 1, 1, 2, 1, 4, 4, 7, 6, 13, 13, 21, 21, 36, 38, 57, 59, 90, 98, 137, 148, 210, 231, 310, 341, 459, 511, 664, 737, 957, 1073, 1357, 1518, 1918, 2156, 2673, 3002, 3712, 4182, 5100, 5737, 6976, 7866, 9460, 10652, 12777, 14402, 17126, 19284, 22867, 25761, 30340, 34139, 40099
OFFSET
0,4
LINKS
FORMULA
G.f.: (1/2) * (Product_{k>=1} (1 - x^(4*k)) / (1 - x^k) - Product_{k>=1} (1 + x^(4*k)) / (1 + x^k)).
a(n) = (A001935(n) - A261734(n)) / 2.
EXAMPLE
a(6) = 4 because we have [6], [3, 2, 1], [2, 2, 2] and [2, 1, 1, 1, 1].
MAPLE
b:= proc(n, i, t) option remember; `if`(n=0, t, `if`(i<1, 0,
b(n, i-1, t)+`if`(irem(i, 4)=0, 0, b(n-i, min(n-i, i), 1-t))))
end:
a:= n-> b(n$2, 0):
seq(a(n), n=0..55); # Alois P. Heinz, Dec 03 2020
MATHEMATICA
nmax = 53; CoefficientList[Series[(1/2) (Product[(1 - x^(4 k))/(1 - x^k), {k, 1, nmax}] - Product[(1 + x^(4 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2020
STATUS
approved