A035625 Number of partitions of n into parts 4k+1 and 4k+3 with at least one part of each type.

0, 0, 0, 1, 1, 1, 2, 4, 4, 5, 7, 10, 12, 14, 18, 24, 28, 33, 41, 50, 59, 70, 84, 100, 117, 137, 161, 188, 219, 254, 295, 341, 393, 453, 520, 595, 682, 780, 889, 1011, 1150, 1307, 1481, 1673, 1893, 2140, 2411, 2713, 3053, 3433, 3851, 4313, 4833, 5411

Offset: 1

Author

Olivier Gérard

Keywords

• nonn

Links

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

Crossrefs

Cf. A035441-A035468, A035618-A035624, A035626-A035699.

Programs

Mathematica

nmax = 54; s1 = Range[0, nmax/4]*4 + 1; s2 = Range[0, nmax/4]*4 + 3;
Table[Count[IntegerPartitions[n, All, s1~Join~s2],
x_ /; ContainsAny[x, s1 ] && ContainsAny[x, s2 ]], {n, 1, nmax}] (* Robert Price, Aug 06 2020 *)
nmax = 54; l = Rest@CoefficientList[Series[(-1 + 1/Product[(1 - x^(4 k + 3)), {k, 0, nmax}])*(-1 + 1/Product[(1 - x^(4 k + 1)), {k, 0, nmax}]), {x, 0, nmax}], x]  (* Robert Price, Aug 16 2020 *)

Formula

G.f.: (-1 + 1/Product_{k>=0} (1 - x^(4 k + 1)))*(-1 + 1/Product_{k>=0} (1 - x^(4 k + 3))). - Robert Price, Aug 16 2020