A246762 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

Offset

0,4

Comments

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

Links

G. C. Greubel, Table of n, a(n) for n = 0..1500

Michael Somos, Introduction to Ramanujan theta functions

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Formula

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^7) * eta(q^28) / (eta(q^2) * eta(q^14))^2 in powers of q.

Euler transform of period 28 sequence [ -1, 1, -1, 0, -1, 1, -2, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, -2, 1, -1, 0, -1, 1, -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 - v^2) * (v - u^2) - 2 * (u*v)^2 * (1 - u*v)^2.

G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).

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

a(n) = (-1)^n * A093950(n).

Convolution inverse of A112212.

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 - 9*x^9 + ...
G.f. = q - q^4 + q^7 - 2*q^10 + 2*q^13 - 3*q^16 + 4*q^19 - 6*q^22 + 7*q^25 + ...

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}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/3)* eta[q]*eta[q^4]*eta[q^7]*eta[q^28]/(eta[q^2]*eta[q^14])^2, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 04 2018 *)

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) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A) / (eta(x^2 + A) * eta(x^14 + A))^2, n))};

Crossrefs

Cf. A093950, A112212.

Sequence in context: A039908 A183954 A266747 * A093950 A373221 A280715

Adjacent sequences: A246759 A246760 A246761 * A246763 A246764 A246765

Keyword

sign

Author

Michael Somos, Sep 02 2014

Status

approved