A035637 -id:A035637 - cp's OEIS frontend

A035636 Number of partitions of n into parts 5k+3 and 5k+4 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 1, 2, 1, 1, 3, 3, 4, 3, 4, 7, 7, 8, 8, 10, 14, 14, 16, 18, 20, 27, 28, 30, 35, 40, 48, 52, 55, 64, 73, 85, 90, 98, 114, 128, 143, 155, 168, 195, 214, 237, 259, 283, 319, 353, 385, 422, 460, 516, 564, 618, 672, 734, 816, 892, 964, 1057

Offset: 1

Olivier Gérard

nonn

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035635, A035637-A035699.

nmax = 66;  s1 = Range[0, nmax/5]*5 + 3; s2 = Range[0, nmax/5]*5 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 07 2020 *)
nmax = 66; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(5 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(5 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(5 k + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(5 k + 4))). - Robert Price, Aug 16 2020

A035638 Number of partitions of n into parts 6k and 6k+2 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 1, 0, 4, 0, 4, 0, 4, 0, 10, 0, 11, 0, 11, 0, 22, 0, 25, 0, 26, 0, 44, 0, 51, 0, 54, 0, 84, 0, 98, 0, 105, 0, 152, 0, 178, 0, 193, 0, 266, 0, 312, 0, 341, 0, 452, 0, 528, 0, 581, 0, 749, 0, 873, 0, 964, 0, 1214, 0, 1409, 0, 1561, 0, 1930, 0, 2234

Offset: 1

Olivier Gérard

nonn

Robert Price, Table of n, a(n) for n = 1..1000

Cf. A035441-A035468, A035618-A035637, A035639-A035699.

nmax = 76; s1 = Range[1, nmax/6]*6; 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 12 2020 *)
nmax = 76; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 2)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

cp's OEIS Frontend

A035636 Number of partitions of n into parts 5k+3 and 5k+4 with at least one part of each type.

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