A260600 Expansion of x * psi(x^3) * psi(x^12) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

0, 1, 1, 2, 4, 6, 9, 14, 20, 29, 42, 58, 80, 111, 149, 200, 268, 353, 463, 606, 784, 1011, 1299, 1656, 2104, 2664, 3354, 4208, 5264, 6555, 8138, 10076, 12428, 15288, 18758, 22944, 27996, 34081, 41377, 50124, 60592, 73075, 87951, 105652, 126652, 151547, 181015

Offset: 0

Author

Michael Somos, Jul 29 2015

Keywords

• nonn

Comments

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

Examples

			G.f. = x + x^2 + 2*x^3 + 4*x^4 + 6*x^5 + 9*x^6 + 14*x^7 + 20*x^8 + ...
G.f. = q^11 + q^17 + 2*q^23 + 4*q^29 + 6*q^35 + 9*q^41 + 14*q^47 + ...
		

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• Vaclav Kotesovec, A method of finding the asymptotics of q-series based on the convolution of generating functions, arXiv:1509.08708 [math.CO], 2015-2016.
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A260574.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(3/2)] EllipticTheta[ 2, 0, x^6] / ( 4 x^(7/8) QPochhammer[ x]), {x, 0, n}];
nmax=60; CoefficientList[Series[x*Product[(1-x^(6*k)) * (1-x^(24*k)) * (1+x^(3*k)) * (1+x^(12*k)) / ((1-x^k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)

PARI

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^6 + A)^2 * eta(x^24 + A)^2 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

PARI

q='q+O('q^99); concat(0, Vec(eta(q^6)^2*eta(q^24)^2 / (eta(q)*eta(q^3)*eta(q^12)))) \\ Altug Alkan, Mar 18 2018

Formula

Expansion of q^(-5/6) * eta(q^6)^2 * eta(q^24)^2 / (eta(q) * eta(q^3) * eta(q^12)) in powers of q.

Euler transform of period 24 sequence [1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 1, 0, 1, 1, 2, 1, 1, -1, ...].

-2 * a(n) = A260574(4*n + 3).

a(n) ~ exp(sqrt(2*n/3)*Pi) / (24*sqrt(2*n)). - Vaclav Kotesovec, Oct 14 2015