A035622 Number of partitions of n into parts 4k and 4k+2 with at least one part of each type.

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 4, 0, 4, 0, 10, 0, 11, 0, 22, 0, 25, 0, 44, 0, 51, 0, 83, 0, 98, 0, 149, 0, 177, 0, 259, 0, 309, 0, 436, 0, 521, 0, 716, 0, 857, 0, 1151, 0, 1376, 0, 1816, 0, 2170, 0, 2818, 0, 3361, 0, 4309, 0, 5132, 0, 6502, 0, 7728, 0, 9695, 0, 11501, 0, 14298

Offset

0,11

Links

Alois P. Heinz, Table of n, a(n) for n = 0..2000 (first 101 terms from Robert Price)

Formula

G.f.: (-1 + 1/Product_{k>=1} (1 - x^(4 k)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 2))). - Robert Price, Aug 16 2020

Mathematica

nmax = 70; s1 = Range[1, nmax/4]*4; s2 = Range[0, nmax/4]*4 + 2;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 0, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 70; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k)), {k, 1, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 2)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020*)

Crossrefs

Bisections give: A006477 (even part), A000004 (odd part).

Cf. A035441-A035468, A035618-A035621, A035623-A035699.

Sequence in context: A256719 A362209 A343722 * A112919 A019201 A137660

Adjacent sequences: A035619 A035620 A035621 * A035623 A035624 A035625

Keyword

nonn

Author

Olivier Gérard

Status

approved