A227317 -id:A227317 - cp's OEIS frontend

A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -10, 37, -50, -30, 128, -25, -34, -320, 310, 410, -370, -87, -410, 320, 30, 500, 384, -630, -640, -359, 300, -326, 2560, -110, -1098, -1280, -370, 1490, -1850, 269, 1500, 1216, 640, 570, -3328, 340, -2010, -1110, 1790, 768, 3200, 303, 750, -1600, -442

Offset: 0

Michael Somos, Sep 02 2013

sign

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

			G.f. = 1 - 10*x + 37*x^2 - 50*x^3 - 30*x^4 + 128*x^5 - 25*x^6 - 34*x^7 - 320*x^8 + ...
G.f. = q - 10*q^5 + 37*q^9 - 50*q^13 - 30*q^17 + 128*q^21 - 25*q^25 - 34*q^29 + ...

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. A000729, A104794, A227317.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x]^5 / QPochhammer[ x^2])^2, {x, 0, n}];

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^5 / eta(x^2 + A))^2, n))};

Expansion of q^(-1/4) * (eta(q)^5 / eta(q^2))^2 in powers of q.
Expansion of phi(-x)^5 * f(-x^2)^3 = phi(-x)^2 * f(-x)^6 in powers of x where phi(), f() are Ramanujan theta functions.
Euler transform of period 2 sequence [ -10, -8, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 8192 (t / i)^4 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A227317.
G.f.: (Product_{k>0} (1 - x^k)^5 / (1 - x^(2*k)))^2.
Convolution of A000729 and A104794.

A228072 Expansion of psi(x^2)^2 * phi(-x^2)^6 + 8 * x * psi(x^2)^6 * phi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, -10, 16, 37, -40, -50, -80, -30, 40, 128, 48, -25, 80, -34, 320, -320, -160, 310, -400, 410, 152, -370, -416, -87, -240, -410, 400, 320, -200, 30, 592, 500, 776, 384, 400, -630, -200, -640, -1120, -359, 552, 300, -272, -326, -800, 2560, -400, -110

Offset: 0

Michael Somos, Sep 02 2013

sign

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

			G.f. = 1 + 8*x - 10*x^2 + 16*x^3 + 37*x^4 - 40*x^5 - 50*x^6 - 80*x^7 - 30*x^8 + ...
G.f. = q + 8*q^3 - 10*q^5 + 16*q^7 + 37*q^9 - 40*q^11 - 50*q^13 - 80*q^15 - 30*q^17 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Hossein Movasati, Younes Nikdelan, Product formulas for weight two newforms, arXiv:1803.01414 [math.NT], 2018.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002171, A227239, A227317, A227695.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2]^12 + 8 x QPochhammer[ x^4]^12) / (QPochhammer[ x^2] QPochhammer[ x^4])^2, {x, 0, n}];

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

Expansion of q^(-1/2) * ((eta(q^2)^5 / eta(q^4))^2 + 8 * (eta(q^4)^5 / eta(q^2))^2) in powers of q.
Expansion of q^(-1/2) * (eta(q^2)^12 + 8 * eta(q^4)^12) / ( eta(q^2) * eta(q^4) )^2 in powers of q.
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(p^e) = b(p) * b(p^(e-1)) - p^3 * b(p^(e-2)) if p>2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 8^2 (t / i)^4 f(t) where q = exp(2 Pi i t).
a(2*n) = A227695(n). a(2*n + 1) = 8 * A227317(n).
If F(x) is the g.f. for A002171, then A(x) * F(x^2) = B(x) the g.f. for A227239. - Michael Somos, Jan 08 2015

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A227695 Expansion of psi(x)^2 * phi(-x)^6 in powers of x where phi(), psi() are Ramanujan theta functions.

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