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A339404
Number of partitions of n into an even number of parts that are not multiples of 3.
4
1, 0, 1, 1, 2, 2, 4, 4, 7, 8, 11, 13, 19, 21, 29, 35, 45, 53, 69, 80, 102, 121, 149, 176, 218, 254, 310, 365, 438, 513, 616, 716, 853, 994, 1172, 1362, 1604, 1853, 2170, 2509, 2920, 3365, 3909, 4488, 5193, 5958, 6862, 7854, 9030, 10303, 11809, 13460, 15376, 17487, 19941, 22624, 25736, 29161
OFFSET
0,5
FORMULA
G.f.: (1/2) * (Product_{k>=1} (1 - x^(3*k)) / (1 - x^k) + Product_{k>=1} (1 + x^(3*k)) / (1 + x^k)).
a(n) = (A000726(n) + A109389(n)) / 2.
EXAMPLE
a(7) = 4 because we have [5, 2], [4, 1, 1, 1], [2, 2, 2, 1] and [2, 1, 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, 3)=0, 0, b(n-i, min(n-i, i), 1-t))))
end:
a:= n-> b(n$2, 1):
seq(a(n), n=0..60); # Alois P. Heinz, Dec 03 2020
MATHEMATICA
nmax = 57; CoefficientList[Series[(1/2) (Product[(1 - x^(3 k))/(1 - x^k), {k, 1, nmax}] + Product[(1 + x^(3 k))/(1 + x^k), {k, 1, nmax}]), {x, 0, nmax}], x]
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Dec 03 2020
STATUS
approved