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A035624
Number of partitions of n into parts 4k+1 and 4k+2 with at least one part of each type.
3
0, 0, 1, 1, 2, 2, 5, 5, 8, 8, 14, 15, 22, 23, 34, 37, 51, 54, 74, 81, 107, 116, 150, 165, 210, 229, 287, 316, 392, 430, 526, 580, 704, 774, 929, 1024, 1223, 1347, 1593, 1756, 2068, 2278, 2663, 2933, 3416, 3762, 4355, 4793, 5529, 6084, 6985, 7680, 8789
OFFSET
1,5
LINKS
Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)
FORMULA
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(4 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 2))). - Robert Price, Aug 16 2020
MATHEMATICA
nmax = 53; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 53; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
KEYWORD
nonn
STATUS
approved