A035649 Number of partitions of n into parts 6k+3 and 6k+4 with at least one part of each type.
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
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 125 terms from Robert Price)
Programs
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Maple
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): seq(a(n), n=1..75); # Alois P. Heinz, Aug 14 2020 -
Mathematica
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 *)
Formula
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