A328802 -id:A328802 - cp's OEIS frontend

A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

1, 2, 1, 0, 0, 2, 2, 0, 2, 2, 1, 0, 2, 6, 2, 0, 3, 6, 4, 0, 4, 8, 4, 0, 7, 14, 7, 0, 6, 16, 10, 0, 11, 20, 11, 0, 14, 32, 16, 0, 17, 38, 21, 0, 22, 46, 24, 0, 32, 66, 34, 0, 34, 78, 44, 0, 49, 96, 50, 0, 60, 130, 66, 0, 72, 154, 84, 0, 90, 186, 98, 0, 117, 244

Offset: 0

Michael Somos, Oct 28 2019

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square of A328802.
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A328789.

			G.f. = 1 + 2*x + x^2 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q^-1 + 2*q^2 + q^5 + 2*q^14 + 2*q^17 + 2*q^23 + 2*q^26 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A112206, A328789, A328790, A328797, A328798, A328802.

a[ n_] := SeriesCoefficient[ (QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6])^2, {x, 0, n}];

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

Euler transform of period 12 sequence [2, -2, 0, 0, 2, -2, 2, 0, 0, -2, 2, 0, ...].
G.f.: Product_{k>=1} (1 + x^(2*k-1))^2 * (1 - x^(6*k-3))^2.
a(n) = (-1)^n * A328797(n). a(2*n) = A112206(n).
a(4*n) = A328789(n). a(4*n + 1) = 2 * A328798(n). a(4*n + 2) = A328790(n). a(4*n + 3) = 0.

A328800 Expansion of chi(-x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 1, -1, 0, 0, 2, -2, 0, 0, 1, -2, 0, 0, 2, -2, 0, 0, 3, -3, 0, 0, 3, -3, 0, 0, 3, -3, 0, 0, 5, -5, 0, 0, 4, -5, 0, 0, 6, -5, 0, 0, 7, -7, 0, 0, 7, -8, 0, 0, 8, -8, 0, 0, 11, -11, 0, 0, 10, -12, 0

Offset: 0

Michael Somos, Oct 27 2019

sign

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution square is A328797.
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A328796.

			G.f. = 1 - x - x^5 + x^8 + x^12 - x^13 + x^16 - x^17 + x^20 + ...
G.f. = q^-1 - q^5 - q^29 + q^47 + q^71 - q^77 + q^95 - q^101 + ...

Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A097242, A328796, A328797, A328802.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ -x^3, x^6], {x, 0, n}];

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

Expansion of q^(1/6) * (eta(q) * eta(q^6)^2) / (eta(q^2) * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-1, 0, 0, 0, -1, -1, -1, 0, 0, 0, -1, 0, ...].
G.f.: Product_{k>=1} (1 - x^(2*k-1)) * (1 + x^(6*k-3)).
a(n) = (-1)^n * A328802. a(4*n) = A097242(n). a(4*n + 1) = -A328796(n). a(4*n + 2) = a(4*n + 3) = 0.

cp's OEIS Frontend

A328795 Expansion of (chi(x) * chi(-x^3))^2 in powers of x where chi() is a Ramanujan theta function.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula