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A035649
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Number of partitions of n into parts 6k+3 and 6k+4 with at least one part of each type.
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3
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0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 3, 1, 1, 3, 3, 1, 7, 3, 3, 8, 8, 3, 14, 9, 8, 16, 17, 9, 27, 19, 18, 32, 34, 20, 49, 40, 37, 58, 63, 43, 87, 74, 70, 104, 113, 82, 149, 135, 128, 177, 195, 152, 249, 232, 224, 298, 327, 266, 407, 392, 380, 485, 535, 455, 654, 639, 628
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OFFSET
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1,13
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LINKS
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FORMULA
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G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 4))). - Robert Price, Aug 16 2020
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MAPLE
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b:= proc(n, i, t, s) option remember; `if`(n=0, t*s, `if`(i<1, 0,
b(n, i-1, t, s)+(h-> `if`(h in {3, 4}, add(b(n-i*j, i-1,
`if`(h=3, 1, t), `if`(h=4, 1, s)), j=1..n/i), 0))(irem(i, 6))))
end:
a:= n-> b(n$2, 0$2):
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MATHEMATICA
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nmax = 69; s1 = Range[0, nmax/6]*6 + 3; s2 = Range[0, nmax/6]*6 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 69; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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