A069911 -id:A069911 - cp's OEIS frontend

A218173 Expansion of f(x^7, x^17) - x^2 * f(x, x^23) in powers of x where f(,) is Ramanujan's two-variable theta function.

1, 0, -1, -1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0

Offset: 0

Michael Somos, Oct 22 2012

sign

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

This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^5, b = x^3.

			1 - x^2 - x^3 + x^7 + x^17 - x^25 - x^28 + x^38 + x^58 - x^72 - x^77 + x^93 + ...
q^25 - q^121 - q^169 + q^361 + q^841 - q^1225 - q^1369 + q^1849 + q^2809 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Quintuple Product Identity

Cf. A010815, A069911.

a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 25]], JacobiSymbol[ 6, Sqrt[48 n + 25]], 0]]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] - QPochhammer[ q]) / 2, {q, 0, 2 n + 1}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] - 1) / 2, {q, 0, 2 n + 1}]; (* Michael Somos, Nov 09 2014 *)

{a(n) = local(m); if( issquare( 48*n + 25, &m), kronecker( 6, m), 0)};

{a(n) = local(m); if( n<0, 0, m = 2*n + 1; - polcoeff( eta( x + x * O(x^m)), m))};

Expansion of f(x, x^7) * chi(-x) in powers of x where f(,) is Ramanujan's two-variable theta function and chi() is a Ramanujan theta function.
G.f.: Sum_{k in Z} x^(12*k^2 + 5*k) - x^(12*k^2 + 11*k + 2).
a(n) = -A010815(2*n + 1).

A366104 G.f. ( Chi(sqrt(x))^4 + Chi(-sqrt(x))^4 )/2, where Chi(x) = Product_{k >= 0} 1 + x^(2*k+1) is the g.f. of A000700.

Original entry on oeis.org

1, 6, 17, 38, 84, 172, 325, 594, 1049, 1796, 3005, 4912, 7877, 12430, 19309, 29580, 44766, 66978, 99150, 145374, 211242, 304382, 435194, 617674, 870651, 1219352, 1697283, 2348888, 3232919, 4426546, 6030872, 8177986, 11039633, 14838518, 19862613, 26482878, 35175989, 46552818, 61393694

Offset: 0

Peter Bala, Sep 29 2023

Compare with A224916 with g.f. ( Chi(sqrt(x))^4 - Chi(-sqrt(x))^4 )/(8*sqrt(x)),
A069910 with g.f. ( Chi(sqrt(x)) + Chi(-sqrt(x)) )/2,
A069911 with g.f. ( Chi(sqrt(x)) - Chi(-sqrt(x)) )/2,
A226622 with g.f. ( Chi(sqrt(x))^2 + Chi(-sqrt(x))^2 )/2 and
A226635 with g.f. ( Chi(sqrt(x))^2 - Chi(-sqrt(x))^2 )/(4*sqrt(x)),
Jacobi's "aequatio identica satis abstrusa" is the identity ( Chi(sqrt(x))^8 - Chi(-sqrt(x))^8 )/(16*sqrt(x)) =  Product_{k >= 1} (1 + x^k)^8.

Cf. A000700, A014705, A069910, A069911, A224916, A226622, A226635.

with(QDifferenceEquations):
 seq(coeff((1/2)*expand(QPochhammer(-q,q^2,40)^4 + QPochhammer(q,q^2,40)^4), q, 2*n), n = 0..40);
#alternative program
seq(coeff(expand(QPochhammer(-q^2, q^2, 20)^2 * QPochhammer(-q, q^2, 20)^6), q, n), n = 0..40);

nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k))^2 * (1 + x^(2*k-1))^6, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Jun 29 2025 *)

G.f.: Product_{k >= 1} (1 + x^(2*k))^2*(1 + x^(2*k-1))^6.
G.f.: x^(1/12) * eta(x^2)^10 * eta(x^4)^2 / ( eta(x) * eta(x^4) )^6.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Jun 29 2025

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A218173 Expansion of f(x^7, x^17) - x^2 * f(x, x^23) in powers of x where f(,) is Ramanujan's two-variable theta function.

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