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

1,5

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^(6 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 2))). - Robert Price, Aug 16 2020

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 *)

Crossrefs

Cf. A035441-A035468, A035618-A035641, A035643-A035699.

Sequence in context: A357415 A038716 A168659 * A213332 A133392 A101199

Adjacent sequences: A035639 A035640 A035641 * A035643 A035644 A035645

Keyword

nonn

Author

Olivier Gérard

Status

approved