A153155 Coefficients of the eighth-order mock theta function T_0(q).

0, 0, 1, -1, 1, -1, 2, -2, 3, -4, 4, -5, 7, -7, 9, -11, 12, -15, 18, -20, 24, -28, 32, -37, 43, -48, 56, -65, 72, -83, 95, -106, 122, -138, 154, -174, 197, -220, 247, -278, 309, -346, 388, -430, 480, -535, 592, -659, 732, -808, 896, -992, 1094, -1209, 1335

Offset

0,7

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

G.f.: Sum{n >= 0} q^((n+1)(n+2)) (1+q^2)(1+q^4)...(1+q^(2n))/(1+q)(1+q^3)...(1+q^(2n+1)).

a(n) ~ (-1)^n * exp(Pi*sqrt(n)/2) / (2^(11/4) * sqrt(1 + sqrt(2)) * sqrt(n)). - Vaclav Kotesovec, Jun 14 2019

Mathematica

nmax = 100; CoefficientList[Series[Sum[x^((k+1)*(k+2)) * Product[(1 + x^(2*j)), {j, 1, k}] / Product[(1 + x^(2*j+1)), {j, 0, k}], {k, 0, Floor[Sqrt[nmax]]}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 13 2019 *)

Prog

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

Crossrefs

Other '8th-order' mock theta functions are at A153148, A153149, A153156, A153172, A153174, A153176, A153178.

Sequence in context: A320470 A320382 A259200 * A225085 A134310 A308663

Adjacent sequences: A153152 A153153 A153154 * A153156 A153157 A153158

Keyword

sign

Author

Jeremy Lovejoy, Dec 19 2008

Extensions

More terms from Michel Marcus, Feb 23 2015

Status

approved