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

1,4

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.

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)

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 *)

Prog

(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

Crossrefs

Cf. A000041, A053274, A328989.

Sequence in context: A092689 A281553 A064434 * A086401 A095732 A001644

Adjacent sequences: A328985 A328986 A328987 * A328989 A328990 A328991

Keyword

nonn

Author

N. J. A. Sloane, Nov 09 2019

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

Status

approved