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

1,10

Links

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

Formula

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

Mathematica

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 *)

Crossrefs

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

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

Sequence in context: A366543 A286663 A114402 * A357487 A225437 A065806

Adjacent sequences: A035644 A035645 A035646 * A035648 A035649 A035650

Keyword

nonn

Author

Olivier Gérard

Status

approved