A153178 Coefficients of the eighth-order mock theta function V_1(q).

0, 1, 1, 1, 2, 3, 3, 4, 5, 6, 8, 9, 11, 14, 16, 19, 23, 27, 31, 37, 43, 49, 58, 66, 76, 89, 101, 115, 132, 150, 170, 194, 219, 247, 280, 315, 354, 399, 447, 500, 562, 627, 699, 781, 869, 967, 1076, 1194, 1323, 1468, 1625, 1796, 1987, 2193, 2418, 2668, 2937, 3231

Offset

0,5

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000

B. Gordon and R. J. McIntosh, Some eighth order mock theta functions, J. London Math. Soc. 62 (2000), 321-335.

Formula

V_1(q) = Sum_{n >= 0} q^((n+1)^2)(1+q)(1+q^3)...(1+q^(2n-1))/((1-q)(1-q^3)...(1-q^(2n+1))).

a(n) ~ exp(Pi*sqrt(n)/2) / (2^(5/2) * sqrt(n)). - Vaclav Kotesovec, Jun 12 2019

Mathematica

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

Prog

(PARI) lista(nn) = {my(q = qq + O(qq^nn)); gf = sum(n = 0, nn, q^((n+1)^2) * prod(k = 1, n, 1 + q^(2*k-1)) / prod(k = 0, n, 1 - q^(2*k+1))); for (i=0, nn-1, print1(polcoeff(gf, i), ", "); ); } \\ Michel Marcus, Jun 18 2013

Crossrefs

Other '8th-order' mock theta functions are at A153148, A153149, A153155, A153156, A153172, A153174, A153176.

Sequence in context: A017864 A188937 A029035 * A118302 A355358 A205216

Adjacent sequences: A153175 A153176 A153177 * A153179 A153180 A153181

Keyword

nonn

Author

Jeremy Lovejoy, Dec 20 2008

Extensions

More terms from Michel Marcus, Feb 23 2015

Status

approved