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

1,5

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.

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)

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 *)

Prog

(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

Crossrefs

Cf. A000041, A053274, A328988.

Sequence in context: A330249 A075550 A292729 * A339190 A137529 A245258

Adjacent sequences: A328986 A328987 A328988 * A328990 A328991 A328992

Keyword

nonn

Author

N. J. A. Sloane, Nov 09 2019

Extensions

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

Status

approved