A233673 -id:A233673 - cp's OEIS frontend

A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

1, -4, 12, -28, 60, -120, 228, -416, 732, -1252, 2088, -3408, 5460, -8600, 13344, -20424, 30876, -46152, 68268, -100016, 145224, -209120, 298800, -423840, 597108, -835804, 1162824, -1608508, 2212896, -3028632, 4124664, -5590976, 7544604, -10137264, 13565016

Offset: 0

Michael Somos, Aug 14 2015

sign

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

			G.f. = 1 - 4*x + 12*x^2 - 28*x^3 + 60*x^4 - 120*x^5 + 228*x^6 + ...

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. A132002, A186924, A260215, A233673.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^3] / EllipticTheta[ 3, 0, q])^2, {q, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^4 + A)^4 * eta(x^6 + A)^10 / (eta(x^2 + A)^10 * eta(x^3 + A)^4 * eta(x^12 + A)^4), n))};

Expansion of eta(q)^4 * eta(q^4)^4 * eta(q^6)^10 / ( eta(q^2)^10 * eta(q^3)^4 * eta(q^12)^4) in powers of q.
Euler transform of period 12 sequence [ -4, 6, 0, 2, -4, 0, -4, 2, 0, 6, -4, 0, ...].
G.f.: (Sum_{k in Z} x^(3*k^2)) / (Sum_{k in Z} x^k^2)^2.
G.f.: (Product_{k>0} (1 + (-x)^k + x^(2*k)) / (1 - (-x)^k + x^(2*k)))^2.
a(n) = (-1)^n * A186924(n) = A233673(3*n) = A260215(3*n).
Convolution square of A132002.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (2*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A261325 Expansion of f(x^3, x^3) * f(x, x^5) / f(x, x)^2 in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

1, -3, 8, -18, 38, -75, 140, -252, 439, -744, 1232, -1998, 3182, -4986, 7700, -11736, 17673, -26322, 38808, -56682, 82070, -117867, 167996, -237744, 334202, -466836, 648224, -895014, 1229148, -1679436, 2283568, -3090672, 4164578, -5587941, 7467464, -9940482

Offset: 0

Michael Somos, Aug 14 2015

sign

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

			G.f. = 1 - 3*x + 8*x^2 - 18*x^3 + 38*x^4 - 75*x^5 + 140*x^6 - 252*x^7 + ...
G.f. = q - 3*q^4 + 8*q^7 - 18*q^10 + 38*q^13 - 75*q^16 + 140*q^19 + ...

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. A187153, A213265, A233670, A233672, A233673, A233698, A260215.

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 *  eta(x^6 + A)^4 / (eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)), n))};

Expansion of f(-x^2) * f(x^3) * f(-x^6) / f(x)^3 in powers of x where f() is a Ramanujan theta function.
Expansion of q^(-1/3) * eta(q)^3 * eta(q^4)^3 *  eta(q^6)^4 / (eta(q^2)^8 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 5, -2, 2, -3, 2, -3, 2, -2, 5, -3, 0, ...].
a(n) = A187153(3*n + 1) = A213265(3*n + 1) = A233670(3*n + 1) = A233672(3*n + 1).
2 * a(n) = A233673(3*n + 1) = - A260215(3*n + 1). a(2*n + 1) = -3 * A233698(n).
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (4*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

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A261320 Expansion of (phi(q^3) / phi(q))^2 in powers of q where phi() is a Ramanujan theta function.

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