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A328989 Number of partitions of n with rank congruent to 1 mod 3. 2
0, 1, 1, 1, 3, 4, 4, 8, 10, 13, 20, 26, 32, 46, 59, 75, 101, 129, 161, 211, 264, 331, 421, 526, 649, 815, 1004, 1235, 1526, 1869, 2275, 2787, 3382, 4097, 4967, 5994, 7205, 8678, 10396, 12437, 14869, 17727, 21076, 25067, 29713, 35174, 41596, 49094, 57827, 68087 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,5

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 3.

FORMULA

a(n) = (A000041(n) - A328988(n))/2. - Alois P. Heinz, Nov 11 2019

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(2-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[2 - i, 3]], {i, 1, n}];

Array[a, 60] (* Jean-François Alcover, Feb 29 2020, after Alois P. Heinz *)

CROSSREFS

Cf. A000041, A328988.

Sequence in context: A330249 A075550 A292729 * A137529 A245258 A086180

Adjacent sequences:  A328986 A328987 A328988 * A328990 A328991 A328992

KEYWORD

nonn

AUTHOR

N. J. A. Sloane, Nov 09 2019

EXTENSIONS

a(22)-a(50) from Lars Blomberg, Nov 11 2019

STATUS

approved

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Last modified August 9 22:52 EDT 2020. Contains 336335 sequences. (Running on oeis4.)