A035620 -id:A035620 - cp's OEIS frontend

A035619 Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.

0, 0, 0, 0, 1, 0, 1, 3, 1, 3, 7, 3, 8, 14, 8, 17, 26, 18, 33, 47, 36, 61, 81, 68, 106, 137, 121, 181, 224, 209, 296, 362, 347, 478, 570, 565, 750, 890, 894, 1166, 1360, 1396, 1774, 2062, 2134, 2677, 3076, 3228, 3973, 4555, 4804, 5854, 6657, 7085, 8513

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 80 terms from Robert Price)

Cf. A035441-A035468, A035618, A035620-A035699.

nmax = 55; s1 = Range[1, nmax/3]*3; s2 = Range[0, nmax/3]*3 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 55; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(3 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(3 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

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

A035621 Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 1, 1, 4, 4, 4, 4, 10, 11, 11, 11, 22, 25, 26, 26, 44, 51, 54, 55, 84, 98, 105, 108, 153, 178, 193, 200, 269, 313, 341, 356, 459, 531, 582, 611, 764, 880, 967, 1021, 1244, 1424, 1568, 1662, 1988, 2264, 2494, 2653, 3122, 3536, 3896, 4155

Offset: 1

Olivier Gérard

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 terms from Robert Price)

Cf. A035441-A035468, A035618-A035620, A035622-A035699.

nmax = 56; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 1;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 56; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

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

A035647 Number of partitions of n into parts 6k+2 and 6k+4 with at least one part of each type.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 4, 0, 5, 0, 7, 0, 11, 0, 14, 0, 19, 0, 26, 0, 33, 0, 43, 0, 55, 0, 70, 0, 88, 0, 111, 0, 137, 0, 170, 0, 208, 0, 256, 0, 311, 0, 378, 0, 456, 0, 551, 0, 658, 0, 790, 0, 940, 0, 1119, 0, 1325, 0, 1570, 0, 1847, 0, 2179, 0, 2554, 0, 2996, 0, 3499

Offset: 1

Olivier Gérard

nonn

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

Cf. A035441-A035468, A035618-A035646, A035648-A035699.

Bisections give A035620 (even part), A000004 (odd part).

nmax = 74; s1 = Range[0, nmax/6]*6 + 2; 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 13 2020 *)
nmax = 74; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 2)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(6 k + 4)), {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 + 4))). - Robert Price, Aug 16 2020

cp's OEIS Frontend

A035619 Number of partitions of n into parts 3k and 3k+2 with at least one part of each type.

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A035621 Number of partitions of n into parts 4k and 4k+1 with at least one part of each type.

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A035647 Number of partitions of n into parts 6k+2 and 6k+4 with at least one part of each type.

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