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

%I A328989 #28 May 23 2023 10:47:23
%S A328989 0,1,1,1,3,4,4,8,10,13,20,26,32,46,59,75,101,129,161,211,264,331,421,
%T A328989 526,649,815,1004,1235,1526,1869,2275,2787,3382,4097,4967,5994,7205,
%U A328989 8678,10396,12437,14869,17727,21076,25067,29713,35174,41596,49094,57827,68087
%N A328989 Number of partitions of n with rank congruent to 1 mod 3.
%C A328989 Also number of partitions of n with rank congruent to 2 mod 3. - _Seiichi Manyama_, May 23 2023
%H A328989 Alois P. Heinz, <a href="/A328989/b328989.txt">Table of n, a(n) for n = 1..10000</a>
%H A328989 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 3.
%F A328989 a(n) = (A000041(n) - A328988(n))/2. - _Alois P. Heinz_, Nov 11 2019
%F A328989 From _Seiichi Manyama_, May 23 2023: (Start)
%F A328989 a(n) = (A000041(n) - A053274(n))/3.
%F A328989 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)
%p A328989 b:= proc(n, i, r) option remember; `if`(n=0 or i=1,
%p A328989       `if`(irem(r+n, 3)=0, 1, 0), b(n, i-1, r)+
%p A328989         b(n-i, min(n-i, i), irem(r+1, 3)))
%p A328989     end:
%p A328989 a:= proc(n) option remember; add(
%p A328989       b(n-i, min(n-i, i), modp(2-i, 3)), i=1..n)
%p A328989     end:
%p A328989 seq(a(n), n=1..60);  # _Alois P. Heinz_, Nov 11 2019
%t A328989 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 A328989 a[n_] := a[n] = Sum[b[n - i, Min[n - i, i], Mod[2 - i, 3]], {i, 1, n}];
%t A328989 Array[a, 60] (* _Jean-François Alcover_, Feb 29 2020, after _Alois P. Heinz_ *)
%o A328989 (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
%Y A328989 Cf. A000041, A053274, A328988.
%K A328989 nonn
%O A328989 1,5
%A A328989 _N. J. A. Sloane_, Nov 09 2019
%E A328989 a(22)-a(50) from _Lars Blomberg_, Nov 11 2019