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A153155 Coefficients of the eighth-order mock theta function T_0(q). 8

%I #16 Jan 31 2021 20:39:19

%S 0,0,1,-1,1,-1,2,-2,3,-4,4,-5,7,-7,9,-11,12,-15,18,-20,24,-28,32,-37,

%T 43,-48,56,-65,72,-83,95,-106,122,-138,154,-174,197,-220,247,-278,309,

%U -346,388,-430,480,-535,592,-659,732,-808,896,-992,1094,-1209,1335

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

%H Vaclav Kotesovec, <a href="/A153155/b153155.txt">Table of n, a(n) for n = 0..10000</a>

%H B. Gordon and R. J. McIntosh, <a href="http://dx.doi.org/10.1112/S0024610700008735">Some eighth order mock theta functions</a>, J. London Math. Soc. 62 (2000), 321-335.

%F 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)).

%F a(n) ~ (-1)^n * exp(Pi*sqrt(n)/2) / (2^(11/4) * sqrt(1 + sqrt(2)) * sqrt(n)). - _Vaclav Kotesovec_, Jun 14 2019

%t 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 *)

%o (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

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

%K sign

%O 0,7

%A _Jeremy Lovejoy_, Dec 19 2008

%E More terms from _Michel Marcus_, Feb 23 2015

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Last modified March 28 05:39 EDT 2024. Contains 371235 sequences. (Running on oeis4.)