A053267 Coefficients of the '5th-order' mock theta function Psi(q).

0, 0, 1, 0, 1, 1, 1, 1, 2, 1, 2, 2, 3, 2, 4, 3, 4, 4, 5, 5, 7, 6, 8, 8, 9, 9, 12, 11, 14, 14, 16, 16, 20, 19, 23, 24, 27, 27, 32, 32, 37, 38, 43, 44, 51, 51, 58, 61, 67, 69, 78, 80, 89, 93, 102, 106, 118, 121, 134, 140, 153, 159, 175, 181, 198, 207, 224, 234, 256, 265, 288

Offset

0,9

References

Dean Hickerson, A proof of the mock theta conjectures, Inventiones Mathematicae, 94 (1988) 639-660

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, pp. 18, 20

Links

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

George E. Andrews and Frank G. Garvan, Ramanujan's "lost" notebook VI: The mock theta conjectures, Advances in Mathematics, 73 (1989) 242-255.

Formula

G.f.: Psi(q) = -1 + Sum_{n>=0} q^(5n^2)/((1-q^2)(1-q^3)(1-q^7)(1-q^8)...(1-q^(5n+2))).

a(n) ~ exp(Pi*sqrt(2*n/15)) / (5^(3/4)*sqrt(2*phi*n)), where phi = A001622 = (1+sqrt(5))/2 is the golden ratio. - Vaclav Kotesovec, Jun 12 2019

Mathematica

Series[Sum[q^(5n^2)/Product[1-q^Abs[5k+2], {k, -n, n}], {n, 0, 4}], {q, 0, 100}]-1
nmax = 100; CoefficientList[Series[-1 + Sum[x^(5*k^2)/ Product[1-x^Abs[5*j+2], {j, -k, k}], {k, 0, Floor[Sqrt[nmax/5]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 12 2019 *)

Crossrefs

Other '5th-order' mock theta functions are at A053256, A053257, A053258, A053259, A053260, A053261, A053262, A053263, A053264, A053265, A053266.

Sequence in context: A051275 A025799 A282537 * A058768 A127682 A127685

Adjacent sequences: A053264 A053265 A053266 * A053268 A053269 A053270

Keyword

nonn,easy

Author

Dean Hickerson, Dec 19 1999

Status

approved