A380481 Number of partitions of n into distinct parts less than n and not a multiple of 3.

1, 0, 0, 1, 0, 1, 2, 2, 2, 3, 3, 4, 6, 6, 7, 9, 9, 11, 14, 15, 17, 20, 22, 25, 30, 33, 37, 42, 46, 52, 60, 66, 73, 82, 90, 101, 114, 125, 138, 153, 168, 186, 207, 227, 249, 274, 300, 330, 364, 398, 435, 476, 519, 568, 622, 678, 738, 804, 874, 952, 1038, 1127, 1223, 1327, 1438, 1561, 1694, 1834, 1984, 2146, 2320, 2509, 2714, 2930, 3161

Offset

0,7

Links

Table of n, a(n) for n=0..74.

Harmandeep Kaur and Muhammad Asif Rana, Partitions with unique largest part and their generating functions, arXiv:2506.11447 [math.CO], 2025. See p. 7.

Formula

G.f.: x^3/(1-x^3) - x/(1-x) + Product_{n>=0} (1+x^(3n+1))*(1+x^(3n+2)).

a(n) ~ exp(Pi*sqrt(2*n)/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Jul 25 2025

a(n) = A003105(n) - A011655(n). - Alois P. Heinz, Jul 25 2025

Example

a(3) = 1: [2,1].
a(5) = 1: [4,1].
a(6) = 2: [5,1], [4,2].
a(7) = 2: [5,2], [4,2,1].
a(8) = 2: [7,1], [5,2,1].
a(9) = 3: [8,1], [7,2], [5,4].
a(10) = 3: [8,2], [7,2,1], [5,4,1].
a(11) = 4: [10,1], [8,2,1], [7,4], [5,4,2].

Maple

b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i<1, 0,
      b(n, i-1)+`if`(irem(i, 3)=0, 0, b(n-i, min(n-i, i-1)))))
    end:
a:= n-> b(n, n-1):
seq(a(n), n=0..74);  # Alois P. Heinz, Jul 24 2025

Mathematica

CoefficientList[
 Series[-q/QPochhammer[q, q, 1] + q^3/QPochhammer[q^3, q, 1] +
    QPochhammer[-q, q^3]*QPochhammer[-q^2, q^3], {q, 0, 500}], q]

Crossrefs

Cf. A003105, A011655, A357457.

Sequence in context: A123576 A094824 A029054 * A341452 A008997 A341453

Adjacent sequences: A380478 A380479 A380480 * A380482 A380483 A380484

Keyword

nonn

Author

Harman Kaur, Jun 23 2025

Status

approved