A093950 Expansion of 1 / (chi(-x) * chi(-x^7)) in powers of x where chi() is a Ramanujan theta function.

1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5962, 6736, 7606, 8582, 9666, 10884

Offset: 0

Author

Michael Somos, Apr 19 2004

Keywords

• nonn

Comments

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

Given g.f. A(x), the right side of Cayley's identity is 2 * q * A(q^2). - Michael Somos, Dec 03 2013

Proof of Cayley's identity, from Silviu Radu, Mar 13 2015: (Start)

Up to issues of convergence I observe that the identity may be rewritten after substituting q=e^{2 Pi Iz} as:

E(28z)^(-1) x E(14z)^2 x E(7z)^(-1) x E(4z)^(-1) x E(2z)^2 x E(z)^(-1) -E(14z)^(-1) x E(7z) x E(2z)^(-1) x E(z) = 2 E(28z) x E(14z)^(-1) x E(4z) x E(2z)^(-1)

where E(z)= exp( Pi I z/12) Product_{n>=1} (1-e^{2 Pi I z n}) is the Dedekind eta function.

One can further rewrite the above identity by dividing the whole identity by the first term. We obtain:

1-E(28z) x E(14z)^(-3) x E(7z)^2 x E(4z) x E(2z)^(-3) x E(z)^2

-2 E(28z)^2 x E(14z)^(-3) x E(7z) x E(4z)^2 x E(2z)^(-3) x E(z)=0

What is interesting now about this expression is that each term is a modular function for the group Gamma_0(28).

Furthermore, all the terms except the constant term have two poles, therefore the whole left hand side has at most two poles (at the points z=1/14 and z=1/2).

However we check that in the q-expansion the first three coefficients are zero, which implies that the left hand side also has a zero of order at least three at the point infinity (note that z=I x infty transforms into q=0, q=e^(2 Pi iz} ).

It is impossible that a nonzero modular function has more zeros than poles, therefore it is the zero function. This finishes the proof. (End)

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 7*x^8 + ...
G.f. = q + q^4 + q^7 + 2*q^10 + 2*q^13 + 3*q^16 + 4*q^19 + 6*q^22 + ...
		

References

• A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

• G. C. Greubel, Table of n, a(n) for n = 0..1000
• A. Cayley, An elliptic-transcendant identity
• Michael Somos, Introduction to Ramanujan theta functions
• Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Crossrefs

Cf. A102314, A112212, A246762.

Programs

Mathematica

a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^7, x^7], {x, 0, n}];

PARI

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + x^(7*k), 1 + x * O(x^n)), n))};

PARI

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^14 + A) / (eta(x + A) * eta(x^7 + A)), n))};

Formula

Expansion of q^(-1/3) * (eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)) in powers of q.

Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, ...].

Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 2*u*v^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. of A102314. - Michael Somos, Dec 03 2013

G.f.: Product_{k>0} (1 + x^k) * (1 + x^(7*k)).

a(n) = A112212(2*n + 1) = - A102314(2*n + 1). - Michael Somos, Dec 03 2013

Convolution inverse of A102314.

a(n) = (-1)^n * A246762(n). - Michael Somos, Sep 02 2014

a(n) ~ exp(2*Pi*sqrt(2*n/21)) / (2^(7/4) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Extensions

Entry revised by N. J. A. Sloane, Mar 15 2015 (with thanks to Doron Zeilberger)