A053278 - OEIS

%I #25 Jan 31 2021 20:46:33

%S 1,1,1,1,1,1,2,2,2,2,2,2,3,3,4,4,4,4,5,5,6,6,7,7,8,8,9,10,11,11,13,13,

%T 14,15,16,17,19,19,21,22,24,25,28,29,31,32,35,36,40,41,44,46,49,51,56,

%U 58,62,65,69,72,77,80,86,90,95,99,106,110,117,122,130,135,144,149,158

%N A '7th-order' mock theta function.

%H Vaclav Kotesovec, <a href="/A053278/b053278.txt">Table of n, a(n) for n = 0..10000</a> (terms 0..1000 from G. C. Greubel)

%H Dean Hickerson, <a href="https://doi.org/10.1007/BF01394280">On the seventh order mock theta functions</a>, Inventiones Mathematicae, 94 (1988) 661-677.

%F G.f.: g(q, q^7), where g(x, q) = Sum_{n >= 1} q^(n*(n-1))/((1-x)*(1-q/x)* (1-q x)*(1-q^2/x)...(1-q^(n-1)/x)*(1-q^n/x)).

%F a(n) ~ exp(Pi*sqrt(2*n/21)) / (2^(3/2) * sin(Pi/7) * sqrt(7*n)). - _Vaclav Kotesovec_, Jun 14 2019

%t Series[Sum[q^(7n(n-1))/Product[1-q^Abs[7k+1], {k, -n, n-1}], {n, 1, 4}], {q, 0, 100}]

%t nmax = 100; CoefficientList[Series[Sum[x^(7*k*(k-1))/Product[1-x^Abs[7*j+1], {j, -k, k-1}], {k, 1, Floor[Sqrt[nmax/7]]+1}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 14 2019 *)

%Y Other '7th-order' mock theta functions are at A053275, A053276, A053277, A053279, A053280.

%K nonn,easy

%O 0,7

%A _Dean Hickerson_, Dec 19 1999