A035642 Number of partitions of n into parts 6k+1 and 6k+2 with at least one part of each type.
0, 0, 1, 1, 2, 2, 3, 3, 6, 6, 9, 9, 12, 12, 18, 19, 26, 27, 34, 35, 46, 49, 63, 66, 81, 84, 104, 111, 137, 146, 174, 183, 218, 233, 278, 297, 348, 368, 428, 457, 534, 572, 660, 702, 803, 858, 984, 1054, 1201, 1280, 1447, 1545, 1749, 1874, 2112, 2255, 2525
Offset: 1
Keywords
Links
- Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 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 {1, 2}, add(b(n-i*j, i-1, `if`(h=1, 1, t), `if`(h=2, 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 = 57; s1 = Range[0, nmax/6]*6 + 1; s2 = Range[0, nmax/6]*6 + 2; Table[Count[IntegerPartitions[n, All, s1~Join~s2], x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *) nmax = 57; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x] (* Robert Price, Aug 16 2020 *)
Formula
G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 2))). - Robert Price, Aug 16 2020