A261454 - OEIS

%I #17 Mar 12 2021 22:24:48

%S 1,1,8,9,17,25,47,63,106,144,216,296,425,569,807,1064,1449,1905,2551,

%T 3304,4353,5592,7254,9247,11859,14978,19038,23872,30034,37433,46734,

%U 57854,71739,88305,108766,133191,163099,198697,242069,293535,355788,429609,518396

%N Expansion of a(x^2) / f(-x) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function.

%C Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

%C Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

%D Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 6, 1st equation.

%H G. C. Greubel, <a href="/A261454/b261454.txt">Table of n, a(n) for n = 0..1000</a>

%H Michael Somos, <a href="/A010815/a010815.txt">Introduction to Ramanujan theta functions</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RamanujanThetaFunctions.html">Ramanujan Theta Functions</a>

%F Expansion of q^(1/24) * (eta(q^2)^3 + 9 * eta(q^18)^3) / (eta(q) * eta(q^6)) in powers of q.

%F Expansion of phi(x) + 2*phi_{-}(x) in powers of x where phi() and phi_{-}() are 6th-order mock theta functions. [Ramanujan]

%F a(n) = A053268(n) + 2*A153251(n). [Ramanujan]

%F a(n) ~ exp(sqrt(2*n/3)*Pi) / (2^(3/2)*sqrt(3*n)). - _Vaclav Kotesovec_, Jun 15 2019

%e G.f. = 1 + x + 8*x^2 + 9*x^3 + 17*x^4 + 25*x^5 + 47*x^6 + 63*x^7 + ...

%e G.f. = 1/q + q^23 + 8*q^47 + 9*q^71 + 17*q^95 + 25*q^119 + 47*q^143 + ...

%t a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^3 + 9 x^2 QPochhammer[ x^18]^3) / (QPochhammer[ x] QPochhammer[ x^6]), {x, 0, n}];

%t nmax = 50; CoefficientList[Series[Product[(1 + x^k)^3*(1 - x^k)^2/(1 - x^(6*k)), {k, 1, nmax}] + 9*x^2*Product[(1 - x^(18*k))^3/((1 - x^k)*(1 - x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* _Vaclav Kotesovec_, Jun 15 2019 *)

%o (PARI) {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 + 9 * x^2 * eta(x^18 + A)^3) / (eta(x + A) * eta(x^6 + A)), n))};

%Y Cf. A053268, A153251.

%K nonn

%O 0,3

%A _Michael Somos_, Nov 18 2015