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

1,7

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

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 *)

Crossrefs

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

Sequence in context: A111138 A349462 A340761 * A219875 A132128 A280057

Adjacent sequences: A035622 A035623 A035624 * A035626 A035627 A035628

Keyword

nonn

Author

Olivier Gérard

Status

approved