A035647 -id:A035647 - cp's OEIS frontend

A035646 Number of partitions of n into parts 6k+2 and 6k+3 with at least one part of each type.

0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 4, 1, 4, 4, 4, 4, 10, 4, 11, 11, 11, 12, 22, 12, 25, 25, 26, 28, 45, 29, 52, 52, 55, 60, 87, 63, 101, 102, 108, 118, 160, 126, 187, 189, 202, 221, 285, 238, 333, 338, 363, 395, 492, 430, 574, 585, 629, 684, 828, 749, 964, 987, 1061

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000 (first 100 terms from Robert Price)

Cf. A035441-A035468, A035618-A035645, A035647-A035699.

nmax = 63; s1 = Range[0, nmax/6]*6 + 2; s2 = Range[0, nmax/6]*6 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 63; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 3)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 3))). - Robert Price, Aug 16 2020

A035648 Number of partitions of n into parts 6k+2 and 6k+5 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 1, 3, 1, 3, 1, 4, 3, 7, 3, 8, 4, 10, 8, 14, 9, 17, 11, 22, 17, 28, 20, 34, 25, 43, 35, 53, 42, 64, 51, 80, 67, 96, 80, 115, 98, 142, 123, 168, 147, 200, 178, 244, 217, 286, 257, 339, 310, 407, 371, 475, 439, 559, 523, 664, 618, 772, 726

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..5000 (first 100 terms from Robert Price)

Cf. A035441-A035468, A035618-A035647, A035649-A035699.

nmax = 68; s1 = Range[0, nmax/6]*6 + 2; s2 = Range[0, nmax/6]*6 + 5;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 13 2020 *)
nmax = 68; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 5)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 5))). - Robert Price, Aug 16 2020

cp's OEIS Frontend

A035646 Number of partitions of n into parts 6k+2 and 6k+3 with at least one part of each type.

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