A328989 - OEIS

%I #28 May 23 2023 10:47:23

%S 0,1,1,1,3,4,4,8,10,13,20,26,32,46,59,75,101,129,161,211,264,331,421,

%T 526,649,815,1004,1235,1526,1869,2275,2787,3382,4097,4967,5994,7205,

%U 8678,10396,12437,14869,17727,21076,25067,29713,35174,41596,49094,57827,68087

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

%C Also number of partitions of n with rank congruent to 2 mod 3. - _Seiichi Manyama_, May 23 2023

%H Alois P. Heinz, <a href="/A328989/b328989.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 3.

%F a(n) = (A000041(n) - A328988(n))/2. - _Alois P. Heinz_, Nov 11 2019

%F From _Seiichi Manyama_, May 23 2023: (Start)

%F a(n) = (A000041(n) - 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^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(2-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[2 - 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)); 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 Cf. A000041, A053274, A328988.

%K nonn

%O 1,5

%A _N. J. A. Sloane_, Nov 09 2019

%E a(22)-a(50) from _Lars Blomberg_, Nov 11 2019