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

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

1,9

Links

Alois P. Heinz, Table of n, a(n) for n = 1..1000 (first 100 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 + 1))). - Robert Price, Aug 16 2020

Mathematica

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*)

Crossrefs

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

Sequence in context: A013189 A295643 A190718 * A046109 A294246 A107680

Adjacent sequences: A035618 A035619 A035620 * A035622 A035623 A035624

Keyword

nonn

Author

Olivier Gérard

Status

approved