A035657 Number of partitions of n into parts 7k+1 and 7k+2 with at least one part of each type.

0, 0, 1, 1, 2, 2, 3, 3, 4, 6, 7, 9, 10, 12, 13, 15, 19, 22, 27, 30, 35, 38, 43, 50, 57, 67, 74, 85, 92, 103, 115, 130, 148, 165, 185, 202, 223, 246, 273, 306, 337, 376, 408, 449, 488, 539, 594, 654, 720, 784, 855, 928, 1013, 1109, 1210, 1326, 1436, 1563, 1685

Offset

1,5

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 + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(7 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, 7))))
    end:
a:= n-> b(n$2, 0$2):
seq(a(n), n=1..75);  # Alois P. Heinz, Aug 14 2020

Mathematica

nmax = 59; s1 = Range[0, nmax/7]*7 + 1; s2 = Range[0, nmax/7]*7 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 14 2020 *)
nmax = 59; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 1)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Crossrefs

Cf. A035441-A035468, A035618-A035656, A035658-A035699.

Sequence in context: A345163 A369728 A369986 * A018053 A146930 A333524

Adjacent sequences: A035654 A035655 A035656 * A035658 A035659 A035660

Keyword

nonn

Author

Olivier Gérard

Status

approved