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
From Seiichi Manyama, May 23 2023: (Start)
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
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