A260186 -id:A260186 - cp's OEIS frontend

A258591 Expansion of (phi(-x^2) * phi(-x^4)^2 / phi(-x)^3)^2 in powers of x where phi() is a Ramanujan theta function.

1, 12, 80, 400, 1664, 6056, 19904, 60320, 171008, 458428, 1171552, 2872368, 6790656, 15544136, 34568576, 74901984, 158507008, 328277848, 666568592, 1329014992, 2605464320, 5028397952, 9563654976, 17942323424, 33232441344, 60814373780, 110029864416

Offset: 0

Michael Somos, Nov 06 2015

nonn

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

			G.f. = 1 + 12*x + 80*x^2 + 400*x^3 + 1664*x^4 + 6056*x^5 + 19904*x^6 + ...

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

Cf. A260186.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2]^2 EllipticTheta[ 4, 0, x^4]^4 / EllipticTheta[ 4, 0, x]^6, {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 * eta(x^4 + A)^3 / (eta(x + A)^6 * eta(x^8 + A)^2))^2, n))};

Expansion of (eta(q^2)^5 * eta(q^4)^3 / (eta(q)^6 * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ 12, 2, 12, -4, 12, 2, 12, 0, ...].
a(n) = A260186(2*n).

A258593 Expansion of (phi(x^2) * psi(x^2) / phi(-x)^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 46, 208, 805, 2776, 8742, 25584, 70450, 184232, 460832, 1108848, 2578295, 5814992, 12760598, 27317056, 57174768, 117223008, 235818894, 466154416, 906606234, 1736736024, 3280271526, 6114139616, 11255369609, 20478505104, 36849912318, 65619691088

Offset: 0

Michael Somos, Nov 06 2015

nonn

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

			G.f. = 1 + 8*x + 46*x^2 + 208*x^3 + 805*x^4 + 2776*x^5 + 8742*x^6 + ...
G.f. = q + 8*q^3 + 46*q^5 + 208*q^7 + 805*q^9 + 2776*q^11 + 8742*q^13 + ...

Vaclav Kotesovec, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A260186.

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^4 + A)^7 / (eta(x + A)^4 * eta(x^2 + A) * eta(x^8 + A)^2))^2, n))};

Expansion of (f(x^2)^2 / (chi(-x^2) * phi(-x)^2))^2 in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Expansion of q^(-1/2) * (eta(q^4)^7 / (eta(q)^4 * eta(q^2) * eta(q^8)^2))^2 in powers of q.
Euler transform of period 8 sequence [ 8, 10, 8, -4, 8, 10, 8, 0, ...].
-4 * a(n) = A260186(2*n + 1).
a(n) ~ exp(2*Pi*sqrt(n)) / (256*n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

cp's OEIS Frontend

A258591 Expansion of (phi(-x^2) * phi(-x^4)^2 / phi(-x)^3)^2 in powers of x where phi() is a Ramanujan theta function.

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