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%I #44 May 23 2023 10:47:05
%S 1,0,1,3,1,3,7,6,10,16,16,25,37,43,58,81,95,127,168,205,264,340,413,
%T 523,660,806,1002,1248,1513,1866,2292,2775,3379,4116,4949,5989,7227,
%U 8659,10393,12464,14845,17720,21109,25041,29708,35210,41562,49085,57871,68052
%N Number of partitions of n with rank a multiple of 3.
%H Alois P. Heinz, <a href="/A328988/b328988.txt">Table of n, a(n) for n = 1..10000</a>
%H Elaine Hou and Meena Jagadeesan, <a href="https://arxiv.org/abs/1607.03846">Dyson’s partition ranks and their multiplicative extensions</a>, arXiv:1607.03846 [math.NT], 2016; The Ramanujan Journal 45.3 (2018): 817-839. See Table 2.
%F a(n) = A000041(n) - 2*A328989(n). - _Alois P. Heinz_, Nov 11 2019
%F From _Seiichi Manyama_, May 23 2023: (Start)
%F a(n) = (A000041(n) + 2*A053274(n))/3.
%F 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)
%p b:= proc(n, i, r) option remember; `if`(n=0 or i=1,
%p `if`(irem(r+n, 3)=0, 1, 0), b(n, i-1, r)+
%p b(n-i, min(n-i, i), irem(r+1, 3)))
%p end:
%p a:= proc(n) option remember; add(
%p b(n-i, min(n-i, i), modp(1-i, 3)), i=1..n)
%p end:
%p seq(a(n), n=1..60); # _Alois P. Heinz_, Nov 11 2019
%t 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]]];
%t a[n_] := a[n] = Sum[b[n - i, Min[n - i, i], Mod[1 - i, 3]], {i, 1, n}];
%t Array[a, 60] (* _Jean-François Alcover_, Feb 29 2020, after _Alois P. Heinz_ *)
%o (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
%Y Cf. A000041, A053274, A328989.
%K nonn
%O 1,4
%A _N. J. A. Sloane_, Nov 09 2019
%E a(33)-a(50) from _Lars Blomberg_, Nov 11 2019
%E 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
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