A035665 -id:A035665 - cp's OEIS frontend

A035664 Number of partitions of n into parts 7k+2 and 7k+5 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 3, 1, 3, 2, 3, 4, 3, 7, 4, 8, 6, 8, 10, 9, 15, 11, 17, 15, 19, 22, 21, 29, 25, 35, 33, 39, 44, 44, 57, 52, 67, 65, 76, 84, 86, 103, 101, 122, 124, 139, 153, 158, 185, 185, 216, 222, 247, 268, 282, 317, 327, 369, 387, 422, 458, 482

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035663, A035665-A035699.

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, 5}, add(b(n-i*j, i-1,
      `if`(h=2, 1, t), `if`(h=5, 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

nmax = 69; s1 = Range[0, nmax/7]*7 + 2; s2 = Range[0, nmax/7]*7 + 5;
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^(7 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(7 k + 5))). - Robert Price, Aug 16 2020

A035666 Number of partitions of n into parts 7k+3 and 7k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 0, 1, 3, 1, 1, 3, 3, 2, 3, 6, 4, 4, 7, 8, 6, 8, 13, 10, 10, 16, 17, 14, 19, 25, 22, 23, 32, 34, 31, 38, 48, 45, 47, 60, 65, 62, 73, 86, 86, 90, 109, 117, 117, 133, 153, 155, 165, 191, 205, 209, 235, 261, 272, 288, 326, 349, 362, 398, 440, 459

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035665, A035667-A035699.

nmax = 71; s1 = Range[0, nmax/7]*7 + 3; s2 = Range[0, nmax/7]*7 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 15 2020 *)
nmax = 71; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(7 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(7 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 15 2020*)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(7*k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(7*k + 4))). - Robert Price, Aug 15 2020

cp's OEIS Frontend

A035664 Number of partitions of n into parts 7k+2 and 7k+5 with at least one part of each type.

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