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A093950 Expansion of 1 / (chi(-x) * chi(-x^7)) in powers of x where chi() is a Ramanujan theta function. 4
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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) prod_{n=1}^{ infty} (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 has also 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)

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

M. Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

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

EXAMPLE

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 + ...

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}];

PROG

(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))};

CROSSREFS

Cf. A102314, A112212, A246762.

Sequence in context: A183954 A266747 A246762 * A280715 A023894 A285799

Adjacent sequences:  A093947 A093948 A093949 * A093951 A093952 A093953

KEYWORD

nonn

AUTHOR

Michael Somos, Apr 19 2004

EXTENSIONS

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

STATUS

approved

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Last modified September 17 06:52 EDT 2019. Contains 327119 sequences. (Running on oeis4.)