A260057 -id:A260057 - cp's OEIS frontend

A260215 Expansion of chi(-q) * chi(q^9) / (chi(q) * chi(-q^9)) in powers of q where chi() is a Ramanujan theta function.

1, -2, 2, -4, 6, -8, 12, -16, 22, -28, 36, -48, 60, -76, 96, -120, 150, -184, 228, -280, 340, -416, 504, -608, 732, -878, 1052, -1252, 1488, -1768, 2088, -2464, 2902, -3408, 3996, -4672, 5460, -6364, 7400, -8600, 9972, -11544, 13344, -15400, 17752, -20424

Offset: 0

Michael Somos, Aug 13 2015

sign

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

			G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 6*x^4 - 8*x^5 + 12*x^6 - 16*x^7 + 22*x^8 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A128143, A260057, A261154, A261156, A261203, A261230, A261325.

a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q, -q] QPochhammer[ -q^9, q^18] QPochhammer[ -q^9, q^9], {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A)), n))};

Expansion of psi(-q) * psi(q^9) / (psi(q) * psi(-q^9)) in powers of q where psi() is a Ramanujan theta function.
Expansion of eta(q)^2 * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9)^2 * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -2, 1, -2, 0, -2, 1, -2, 0, 0, 1, -2, 0, -2, 1, -2, 0, -2, 0, -2, 0, -2, 1, -2, 0, -2, 1, 0, 0, -2, 1, -2, 0, -2, 1, -2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. of A128143.
a(n) = (-1)^n * A261156(n). Convolution inverse of A261156
a(2*n + 1) = -2 * A261203(n) = -2 * A261154(2*n + 1). 2 * a(2*n) = A261154(2*n) unless n=0.
a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = 2 * A260057(n). - Michael Somos, Nov 08 2015
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A260150 Expansion of f(x, x^5)^3 / (f(-x, -x^5) * f(-x^2, -x^2)^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 4, 11, 24, 48, 92, 170, 304, 526, 884, 1451, 2336, 3700, 5772, 8876, 13472, 20207, 29988, 44072, 64184, 92680, 132760, 188758, 266512, 373838, 521152, 722266, 995432, 1364684, 1861548, 2527224, 3415344, 4595497, 6157700, 8218050, 10925848, 14472520

Offset: 0

Michael Somos, Nov 08 2015

nonn

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

			G.f. = 1 + 4*x + 11*x^2 + 24*x^3 + 48*x^4 + 92*x^5 + 170*x^6 + 304*x^7 + ...
G.f. = q^2 + 4*q^5 + 11*q^8 + 24*q^11 + 48*q^14 + 92*q^17 + 170*q^20 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A259033, A260057, A261154.

a[ n_] := SeriesCoefficient[ 2^(-1/2) x^(-3/4) QPochhammer[ -x] / QPochhammer[x]^3 EllipticTheta[ 2, Pi/4, x^(3/2)]^3 / EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^4 * eta(x^12 + A)^3 / (eta(x + A)^4 * eta(x^4 + A) * eta(x^6 + A)^5), n))};

Expansion of f(x) * psi(-x^3)^3 / (f(-x)^3 * psi(x^3)) in powers of x where psi(), f() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 4, 1, 0, 2, 4, 2, 4, 2, 0, 1, 4, 0, ...].
a(n) = (-1)^n * A260057(n). a(n) = A261154(3*n + 2). a(2*n + 1) = 4 * A259033(n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

cp's OEIS Frontend

A260215 Expansion of chi(-q) * chi(q^9) / (chi(q) * chi(-q^9)) in powers of q where chi() is a Ramanujan theta function.

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