A187149 -id:A187149 - cp's OEIS frontend

A106508 Expansion of psi(x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

1, 4, 4, 0, 2, 0, -8, 0, -5, -16, 4, 0, -10, 0, -8, 0, 9, 8, 0, 0, 14, 0, 16, 0, -10, 32, 4, 0, 0, 0, 8, 0, 14, -20, -20, 0, 2, 0, 0, 0, -11, -16, -20, 0, -32, 0, 16, 0, 0, -40, 4, 0, 14, 0, -8, 0, -9, 32, -20, 0, 26, 0, 0, 0, 2, 36, 28, 0, 0, 0, 16, 0, 16, 0, 28, 0, -22, 0, 0, 0, 14, 56, -16, 0, 0, 0, -40, 0, 0

Offset: 0

Michael Somos, May 24 2005

sign

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

			1 + 4*x + 4*x^2 + 2*x^4 - 8*x^6 - 5*x^8 - 16*x^9 + 4*x^10 - 10*x^12 + ...
q + 4*q^4 + 4*q^7 + 2*q^13 - 8*q^19 - 5*q^25 - 16*q^28 + 4*q^31 - 10*q^37 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
C. Adiga, N. Anitha and T. Kim, Transformations of Ramanujan's Summation Formula and its Applications, arXiv:math/0501528 [math.NT], 2005. See page 5.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A187149.

a[n_]:= SeriesCoefficient[(x^(-1/2)/16)*EllipticTheta[2, 0, x^(1/2)]^4* QPochhammer[x^2, x^4]^2, {x, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Jan 04 2018 *)

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

Expansion of q^(-1/3) * eta(q^2)^10 / (eta(q)^4 * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [4, -6, 4, -4, ...].
a(n) = (-1)^n * A187149(n). a(4*n + 3) = a(8*n + 5) = 0.
G.f. Product_{k>0} (1 + x^k)^4 (1 - x^(2*k))^4 / (1 + x^(2*k))^2.

A187150 Expansion of psi(-x)^4 / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 1, -2, 0, 4, 1, 2, -5, 0, -5, 4, 1, -2, -5, 0, 7, 4, 7, 0, -4, -10, 7, -8, 0, 4, 0, -8, 2, 0, 1, -2, 0, 2, 0, 14, 7, 0, -5, 10, -11, -8, -10, -2, 0, 10, -4, 4, 0, 0, -5, -8, -11, 10, 0, 0, 14, -2, 20, 0, -11, 4, 13, 2, -5, -14, 0, -14, 13, 0, -11, -14, 8, -2, 0, 10, 13, -18, 0, 0, -5

Offset: 0

Michael Somos, Mar 05 2011

sign

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

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 4*x^5 + x^6 + 2*x^7 - 5*x^8 - 5*x^10 + ...
G.f. = q^7 - 2*q^19 + q^31 - 2*q^43 + 4*q^67 + q^79 + 2*q^91 - 5*q^103 + ...

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. A187149.

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2])^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)
a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, Pi/4, x^(1/2)]^4 / QPochhammer[ x, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 02 2015 *)

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

Expansion of q^(-7/12) * (eta(q) * eta(q^4)^2 / eta(q^2))^2 in powers of q.
Euler transform of period 4 sequence [ -2, 0, -2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187149.

A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -5, -10, 9, 14, -10, 0, 14, 2, -11, -32, 0, 14, -9, 26, 2, 0, 16, -22, 14, 0, 0, 26, -17, -32, -22, -10, -34, 14, 45, 38, 0, -34, 38, -22, 2, 0, -10, 64, -20, 0, 0, 0, -23, -46, 16, 0, -46, -32, 26, -10, 25, 18, 0, 38, 50, 0, 0, -22, -80, 50, 0, 26, 2

Offset: 0

Michael Somos, Jun 09 2015

sign

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

			G.f. = 1 + 2*x - 5*x^2 - 10*x^3 + 9*x^4 + 14*x^5 - 10*x^6 + 14*x^8 + ...
G.f. = q + 2*q^13 - 5*q^25 - 10*q^37 + 9*q^49 + 14*q^61 - 10*q^73 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A000727, A106508, A143378, A187076, A187149.

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

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

Expansion of q^(-1/12) * (eta(q^2)^5 / (eta(q) * eta(q^4)^2))^2 in powers of q.
Euler transform of period 4 sequence [ 2, -8, 2, -4, ...].
a(n) = A000727(2*n) = A187076(2*n) = A106508(4*n) = A187149(4*n).
Convolution square of A143378.

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A106508 Expansion of psi(x)^4 * chi(-x^2)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

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A187150 Expansion of psi(-x)^4 / chi(-x)^2 in powers of x where psi(), chi() are Ramanujan theta functions.

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A258779 Expansion of (f(-x) * phi(x))^2 in powers of x where phi(), f() are Ramanujan theta functions.

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