A053269 - OEIS

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

(list; graph; refs; listen; history; text; internal format)

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Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 4, 13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