A035639 -id:A035639 - cp's OEIS frontend

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

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

A035640 Number of partitions of n into parts 6k and 6k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 3, 0, 1, 0, 3, 0, 7, 0, 3, 0, 8, 0, 14, 0, 8, 0, 17, 0, 26, 0, 18, 0, 33, 0, 47, 0, 36, 0, 61, 0, 81, 0, 68, 0, 106, 0, 137, 0, 121, 0, 181, 0, 224, 0, 209, 0, 296, 0, 362, 0, 347, 0, 478, 0, 570, 0, 565, 0, 750, 0, 890, 0, 894, 0, 1166, 0

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035639, A035641-A035699.

Bisections give: A035619 (even part), A000004 (odd part).

nmax = 81; s1 = Range[1, nmax/6]*6; s2 = Range[0, nmax/6]*6 + 4;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 12 2020 *)
nmax = 81; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 12 2020 *)

G.f. : (-1 + 1/Product_{k>=0} (1 - x^(6 k + 4)))*(-1 + 1/Product_{k>=1} (1 - x^(6 k))). - Robert Price, Aug 12 2020

cp's OEIS Frontend

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

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