A132218 Expansion of psi(-x^3) / phi(-x) in powers of x where psi(), phi() are Ramanujan theta functions.

1, 2, 4, 7, 12, 20, 32, 50, 76, 113, 166, 240, 342, 482, 672, 928, 1270, 1724, 2323, 3108, 4132, 5460, 7174, 9376, 12192, 15780, 20332, 26086, 33334, 42432, 53817, 68018, 85680, 107584, 134674, 168092, 209210, 259680, 321484, 396996, 489056, 601052, 737024

Offset: 0

Author

Michael Somos, Aug 13 2007

Keywords

• nonn

Comments

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

Examples

			G.f. = 1 + 2*x + 4*x^2 + 7*x^3 + 12*x^4 + 20*x^5 + 32*x^6 + 50*x^7 + 76*x^8 + ...
G.f. = q^3 + 2*q^11 + 4*q^19 + 7*q^27 + 12*q^35 + 20*q^43 + 32*q^51 + 50*q^59 + ...
		

Links

• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], Sep 30 2015
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Programs

Mathematica

nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(12*k))/( (1-x^k) * (1+x^(3*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/8) EllipticTheta[ 2, Pi/4, x^(3/2)] / EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)

PARI

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of q^(-3/8) * eta(q^2) * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^6)) in powers of q.

Euler transform of period 12 sequence [ 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 0, ...].

G.f.: Product_{k>0} (1 + x^k) * (1 + x^k + x^(2*k)) * (1 + x^(6*k)).

a(n) ~ 5^(1/4) * exp(sqrt(5*n/6)*Pi) / (2^(11/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015