A328989 -id:A328989 - cp's OEIS frontend

A053274 Coefficients of the '6th-order' mock theta function gamma(q).

1, 1, -1, 0, 2, -2, -1, 3, -2, 0, 3, -4, -1, 5, -3, -1, 6, -6, -2, 7, -6, 0, 9, -8, -3, 11, -9, -2, 13, -13, -3, 17, -12, -3, 19, -18, -5, 22, -19, -3, 27, -24, -7, 33, -26, -5, 36, -34, -9, 44, -35, -9, 51, -45, -11, 58, -49, -9, 68, -59, -16, 78, -65, -15, 88, -79, -19, 104, -84, -19, 117, -102, -26, 133, -112, -24, 152, -131

Offset: 0

Dean Hickerson, Dec 19 1999

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17

G. C. Greubel, Table of n, a(n) for n = 0..1000
George E. Andrews and Dean Hickerson, Ramanujan's "lost" notebook VII: The sixth order mock theta functions, Advances in Mathematics, 89 (1991) 60-105.

Other '6th-order' mock theta functions are at A053268, A053269, A053270, A053271, A053272, A053273.

Cf. A328988, A328989.

Series[Sum[q^n^2/Product[1+q^k+q^(2k), {k, 1, n}], {n, 0, 10}], {q, 0, 100}]

a(n) = polcoeff(sum(k=0, 50, q^(k^2)/prod(j=1, k, 1+q^j+q^(2*j)), q*O(q^n)), n);
for(n=0,50, print1(a(n), ", ")) \\ G. C. Greubel, May 18 2018

my(N=80, x='x+O('x^N)); Vec(1+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)^2/(1+x^k+x^(2*k)))) \\ Seiichi Manyama, May 23 2023

G.f.: gamma(q) = Sum_{n >= 0} q^n^2/((1+q+q^2)(1+q^2+q^4)...(1+q^n+q^(2n))).
From Seiichi Manyama, May 23 2023: (Start)
a(n) = A328988(n) - A328989(n) for n > 0.
G.f.: 1 + (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)^2 / (1+x^k+x^(2*k)). (End)

A328988 Number of partitions of n with rank a multiple of 3.

Original entry on oeis.org

1, 0, 1, 3, 1, 3, 7, 6, 10, 16, 16, 25, 37, 43, 58, 81, 95, 127, 168, 205, 264, 340, 413, 523, 660, 806, 1002, 1248, 1513, 1866, 2292, 2775, 3379, 4116, 4949, 5989, 7227, 8659, 10393, 12464, 14845, 17720, 21109, 25041, 29708, 35210, 41562, 49085, 57871, 68052

Offset: 1

N. J. A. Sloane, Nov 09 2019

nonn

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.

Cf. A000041, A053274, A328989.

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

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 *)

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

a(n) = A000041(n) - 2*A328989(n). - Alois P. Heinz, Nov 11 2019
From Seiichi Manyama, May 23 2023: (Start)
a(n) = (A000041(n) + 2*A053274(n))/3.
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)

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

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A053274 Coefficients of the '6th-order' mock theta function gamma(q).

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A328988 Number of partitions of n with rank a multiple of 3.

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