A035648 -id:A035648 - cp's OEIS frontend

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

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

A035649 Number of partitions of n into parts 6k+3 and 6k+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, 3, 1, 1, 3, 3, 1, 7, 3, 3, 8, 8, 3, 14, 9, 8, 16, 17, 9, 27, 19, 18, 32, 34, 20, 49, 40, 37, 58, 63, 43, 87, 74, 70, 104, 113, 82, 149, 135, 128, 177, 195, 152, 249, 232, 224, 298, 327, 266, 407, 392, 380, 485, 535, 455, 654, 639, 628

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-A035648, A035650-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 {3, 4}, add(b(n-i*j, i-1,
      `if`(h=3, 1, t), `if`(h=4, 1, s)), j=1..n/i), 0))(irem(i, 6))))
    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/6]*6 + 3; 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 14 2020 *)
nmax = 69; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(6 k + 3)), {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 + 3)))*(-1 + 1/Product_{k>=0} (1 - x^(6 k + 4))). - Robert Price, Aug 16 2020

cp's OEIS Frontend

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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