A328988 Number of partitions of n with rank a multiple of 3.

1, 0, 1, 3, 1, 3, 7, 6, 10, 16, 16, 25, 37, 43, 58, 81, 95, 127, 168, 205, 264, 340, 413, 523, 660, 806, 1002, 1248, 1513, 1866, 2292, 2775, 3379, 4116, 4949, 5989, 7227, 8659, 10393, 12464, 14845, 17720, 21109, 25041, 29708, 35210, 41562, 49085, 57871, 68052

Offset: 1

Author

N. J. A. Sloane, Nov 09 2019

Keywords

• nonn

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..10000
• Elaine Hou and Meena Jagadeesan, Dyson’s partition ranks and their multiplicative extensions, arXiv:1607.03846 [math.NT], 2016; The Ramanujan Journal 45.3 (2018): 817-839. See Table 2.

Crossrefs

Cf. A000041, A053274, A328989.

Programs

Maple

b:= proc(n, i, r) option remember; `if`(n=0 or i=1,
      `if`(irem(r+n, 3)=0, 1, 0), b(n, i-1, r)+
        b(n-i, min(n-i, i), irem(r+1, 3)))
    end:
a:= proc(n) option remember; add(
      b(n-i, min(n-i, i), modp(1-i, 3)), i=1..n)
    end:
seq(a(n), n=1..60);  # Alois P. Heinz, Nov 11 2019

Mathematica

b[n_, i_, r_] := b[n, i, r] = If[n == 0 || i == 1, If[Mod[r + n, 3] == 0, 1, 0], b[n, i - 1, r] + b[n - i, Min[n - i, i], Mod[r + 1, 3]]];
a[n_] := a[n] = Sum[b[n - i, Min[n - i, i], Mod[1 - i, 3]], {i, 1, n}];
Array[a, 60] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)

PARI

my(N=60, x='x+O('x^N)); Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k-1)/2)*(1+x^(3*k))/(1+x^k+x^(2*k)))) \\ Seiichi Manyama, May 23 2023

Formula

a(n) = A000041(n) - 2*A328989(n). - Alois P. Heinz, Nov 11 2019

From Seiichi Manyama, May 23 2023: (Start)

a(n) = (A000041(n) + 2*A053274(n))/3.

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

Extensions

a(33)-a(50) from Lars Blomberg, Nov 11 2019

Typo in a(14) in both the arXiv preprint and the published version in the Ramanujan Journal corrected by Alois P. Heinz, Nov 11 2019