A328989 Number of partitions of n with rank congruent to 1 mod 3.

0, 1, 1, 1, 3, 4, 4, 8, 10, 13, 20, 26, 32, 46, 59, 75, 101, 129, 161, 211, 264, 331, 421, 526, 649, 815, 1004, 1235, 1526, 1869, 2275, 2787, 3382, 4097, 4967, 5994, 7205, 8678, 10396, 12437, 14869, 17727, 21076, 25067, 29713, 35174, 41596, 49094, 57827, 68087

Offset: 1

Author

N. J. A. Sloane, Nov 09 2019

Keywords

• nonn

Comments

Also number of partitions of n with rank congruent to 2 mod 3. - Seiichi Manyama, May 23 2023

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 3.

Crossrefs

Cf. A000041, A053274, A328988.

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(2-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[2 - 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)); concat(0, Vec(1/prod(k=1, N, 1-x^k)*sum(k=1, N, (-1)^(k-1)*x^(k*(3*k+1)/2)*(1+x^k)/(1+x^k+x^(2*k))))) \\ Seiichi Manyama, May 23 2023

Formula

a(n) = (A000041(n) - A328988(n))/2. - Alois P. Heinz, Nov 11 2019

From Seiichi Manyama, May 23 2023: (Start)

a(n) = (A000041(n) - 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^k) / (1+x^k+x^(2*k)). (End)

Extensions

a(22)-a(50) from Lars Blomberg, Nov 11 2019