A035665 Number of partitions of n into parts 7k+2 and 7k+6 with at least one part of each type.

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 2, 2, 2, 2, 2, 2, 3, 4, 6, 4, 7, 4, 8, 6, 12, 10, 13, 12, 14, 15, 18, 21, 24, 23, 29, 26, 35, 33, 46, 41, 52, 50, 58, 61, 71, 77, 84, 89, 99, 101, 120, 121, 146, 142, 168, 166, 192, 199, 226, 239, 259, 275, 300, 316, 354, 370, 416, 422

Offset

1,14

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 125 terms from Robert Price)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 6))). - 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 {2, 6}, add(b(n-i*j, i-1,
      `if`(h=2, 1, t), `if`(h=6, 1, s)), j=1..n/i), 0))(irem(i, 7))))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75);  # Alois P. Heinz, Aug 14 2020

Mathematica

nmax = 71; s1 = Range[0, nmax/7]*7 + 2; s2 = Range[0, nmax/7]*7 + 6;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 6)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035664, A035666-A035699.

Sequence in context: A241087 A137722 A081305 * A352829 A307833 A331566

Adjacent sequences: A035662 A035663 A035664 * A035666 A035667 A035668

Keyword

nonn

Author

Olivier Gérard

Status

approved