A164270 -id:A164270 - cp's OEIS frontend

A164269 Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.

1, 2, 0, -7, -12, 0, 32, 50, 0, -114, -168, 0, 350, 496, 0, -967, -1332, 0, 2468, 3324, 0, -5916, -7824, 0, 13471, 17548, 0, -29384, -37788, 0, 61784, 78578, 0, -125838, -158496, 0, 249230, 311224, 0, -481506, -596676, 0, 909788, 1119624, 0, -1684824, -2060448, 0

Offset: 1

Michael Somos, Aug 11 2009

sign

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

			G.f. = q + 2*q^2 - 7*q^4 - 12*q^5 + 32*q^7 + 50*q^8 - 114*q^10 - 168*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A164268, A164270, A164271.

f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164269[n_] := SeriesCoefficient[q*(f[q^9, -q^18]^3*f[q, q])/(( f[q^3, -q^6])*f[q^3, q^3]^3), {q, 0, n}]; Rest[Table[A164269[n], {n,0,50}]] (* G. C. Greubel, Sep 16 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^9]^3 EllipticTheta[ 3, 0, x] / (QPochhammer[ -x^3] EllipticTheta[ 3, 0, x^3]^3), {x, 0, n}]; (* Michael Somos, Sep 17 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ x^9]^3 EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^3]^4, {x, 0, n}]; /(* Michael Somos, Sep 20 2017 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3), n))};

Euler transform of period 36 sequence [2, -3, -5, -1, 2, 8, 2, -1, -2, -3, 2, 3, 2, -3, -5, -1, 2, 2, 2, -1, -5, -3, 2, 3, 2, -3, -2, -1, 2, 8, 2, -1, -5, -3, 2, 0, ...].
a(3*n) = 0. a(3*n + 1) = A164270(n). a(3*n + 2) = 2 * A164271(n).
Convolution inverse of A164268.
Expansion of x * phi(x) * chi(x^3) * f(x^9)^3 / phi(x^3)^4 = x * phi(x) * f(x^9)^3 / (chi(x^3)^3 * f(x^3)^4) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 20 2017

A294387 Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 6, -12, 21, -36, 60, -96, 150, -228, 342, -504, 732, -1050, 1488, -2088, 2901, -3996, 5460, -7404, 9972, -13344, 17748, -23472, 30876, -40413, 52644, -68268, 88152, -113364, 145224, -185352, 235734, -298800, 377514, -475488, 597108, -747690, 933672

Offset: 0

Michael Somos, Oct 29 2017

sign

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

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = 1 - 3*x + 6*x^2 - 12*x^3 + 21*x^4 - 36*x^5 + 60*x^6 - 96*x^7 + ...

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. A128111, A128128, A132972, A164270, A164271.

a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q]^3 / QPochhammer[ q^3, -q^3], {q, 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)^2 / (eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^12 + A)), n))};

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3*eta(q^4)^3*eta(q^6)^2/(eta(q^2)^6*eta(q^3)*eta(q^12)))} \\ Altug Alkan, Mar 21 2018

Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6)^2 / (eta(q^2)^6 * eta(q^3) * eta(q^12)) in powers of q.
Expansion of (c(q) - c(q^4)) * (c(q) - 4*c(q^4)) / (c(q) + 2*c(q^4))^2 in powers of q where c(q) is a cubic AGM theta function.
Expansion of b(q^2) / b(-q) = b(q^2) / (2*b(q^4) - b(q)) in powers of q where b() is a cubic AGM theta function.
Expansion of (3*a(q^12) - a(q^4)) / (a(q) + a(q^2)) = -1/2 + 3/2*(a(-q^3) + 2*a(q^3)) / (2*a(q) + a(-q)) in powers of q where a() is a cubic AGM theta function.
Euler transform of period 12 sequence [-3, 3, -2, 0, -3, 2, -3, 0, -2, 3, -3, 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. for A128111.
G.f. A(q) = (1 - T(q)) / (1 + 2*T(q))  where T(q) = q*A128111(q^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v) + 3*(u*v)^2 - 4*(u*v)^3 + 2*(u*v)^4 - (u^3 + v^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u*(1 + u + u^2) - v^3*(1 - 2*u + 4*u^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1 + u2 + u1*u2 - u3*u6 - 2*u1*u2*u3*u6.
G.f.: Product_{k>0} (1 + x^(6*k-3)) / (1 + x^(2*k-1))^3.
a(n) = (-1)^n * A128128(n). Convolution inverse of A132972.
a(3*n + 1) = -3 * A164270(n). a(3*n + 2) = 6 * A164271(n).
Empirical : Sum_{n>=0} a(n)/exp(Pi*n) = 1/2*(2+2*3^(1/2))^(1/3), validated up to 1000 digits. - Simon Plouffe, May 06 2023

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A164269 Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.

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A294387 Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

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