A053269 Coefficients of the '6th-order' mock theta function psi(q).

0, 1, -1, 1, -2, 3, -2, 2, -4, 5, -5, 5, -7, 9, -8, 9, -12, 14, -15, 16, -20, 23, -23, 25, -31, 36, -37, 40, -47, 54, -56, 60, -71, 79, -84, 91, -103, 115, -121, 131, -149, 164, -174, 188, -210, 232, -245, 264, -294, 321, -343, 368, -406, 443, -470, 505, -554, 602, -641, 687, -751, 813, -863, 925

Offset

0,5

References

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 4, 13

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)

George E. Andrews and Dean Hickerson, Ramanujan's "lost" notebook VII: The sixth order mock theta functions, Advances in Mathematics, 89 (1991) 60-105.

Formula

G.f.: psi(q) = Sum_{n >= 0} (-1)^n q^(n+1)^2 (1-q)*(1-q^3)...(1-q^(2n-1)) /((1+q)*(1+q^2)...(1+q^(2n+1))).

a(3*n + 1) = A262614(n). a(3*n + 2) = - A263041(n). - Michael Somos, Apr 17 2016

a(n) ~ -(-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(3*n)). - Vaclav Kotesovec, Jun 15 2019

Mathematica

Series[Sum[(-1)^n q^(n+1)^2 Product[1-q^k, {k, 1, 2n-1, 2}]/Product[1+ q^k, {k, 1, 2n+1}], {n, 0, 9}], {q, 0, 100}]
nmax = 100; CoefficientList[Series[Sum[(-1)^k * x^((k+1)^2) * Product[1-x^j, {j, 1, 2*k-1, 2}]/Product[1+ x^j, {j, 1, 2*k+1}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 15 2019 *)

Crossrefs

Other '6th-order' mock theta functions are at A053268, A053270, A053271, A053272, A053273, A053274.

Cf. A262614, A263041.

Sequence in context: A194020 A372053 A372048 * A163873 A309563 A292588

Adjacent sequences: A053266 A053267 A053268 * A053270 A053271 A053272

Keyword

sign,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved