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

Author

Dean Hickerson, Dec 19 1999

Keywords

• sign
• easy

References

• Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 17

Links

• 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.

Crossrefs

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

Cf. A328988, A328989.

Programs

Mathematica

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

PARI

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

PARI

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

Formula

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)