A029838 -id:A029838 - cp's OEIS frontend

A134178 Expansion of chi(x) * chi(-x^2)^2 * chi(-x^3) * chi(-x^4) * chi(x^6)^2 * chi(-x^12) in powers of x where chi() is a Ramanujan theta function.

1, 1, -2, -2, 0, 1, 2, 0, 0, -1, -4, 0, 1, 0, 6, 2, 0, 1, -8, 0, 0, 0, 12, 0, -1, -1, -18, -4, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 6, 0, -2, -58, 0, 0, -1, 76, 0, 1, 2, -100, -8, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 12, 0, 4, -264, 0, 0, 2, 332, 0, -1, -5, -416, -18, 0, -2, 516, 0, 0, 5, -640, 0, -1, 2, 790, 24

Offset: 0

Michael Somos, Oct 11 2007

sign

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

			G.f. = 1 + x - 2*x^2 - 2*x^3 + x^5 + 2*x^6 - x^9 - 4*x^10 + x^12 + 6*x^14 + ...
G.f. = q^-3 + q^-1 - 2*q - 2*q^3 + q^7 + 2*q^9 - q^15 - 4*q^17 + q^21 + 6*q^25 + ...

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. A029838, A083365.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^2, x^4]^2 QPochhammer[ x^3, x^6] QPochhammer[ x^4, x^8] QPochhammer[-x^6, x^12]^2 QPochhammer[ x^12, x^24], {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^12, x^24] QPochhammer[ x^24, x^48] + x QPochhammer[ -x^4, x^8] QPochhammer[ x^8, x^16] - 2 x^2 QPochhammer[ x^16] / QPochhammer[ -x^4] - 2 x^3 QPochhammer[ x^48] / QPochhammer[ -x^12], {x, 0, n}]; (* Michael Somos, Oct 25 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^12 + A)^5 / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^24 + A)^3), n))};

Euler transform of period 24 sequence [ 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -4, 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, 0, ...].
a(12*n + 4) = a(12*n + 7) = a(12*n + 8) = a(12*n + 11) = 0.
a(4*n + 1) = a(12*n) = A029838(n). a(4*n + 2) = a(12*n + 3) = -2 * A083365(n).

A208589 Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 0, 0, -1, 4, 0, 0, 0, -6, 0, 0, 1, 8, 0, 0, 0, -12, 0, 0, -1, 18, 0, 0, -1, -24, 0, 0, 2, 32, 0, 0, 1, -44, 0, 0, -2, 58, 0, 0, -1, -76, 0, 0, 2, 100, 0, 0, 1, -128, 0, 0, -3, 164, 0, 0, -1, -210, 0, 0, 4, 264, 0, 0, 2, -332, 0, 0, -5, 416

Offset: 0

Michael Somos, Feb 29 2012

sign

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

			G.f. = 1 + 2*x + x^4 - 2*x^5 - x^8 + 4*x^9 - 6*x^13 + x^16 + 8*x^17 - 12*x^21 + ...
G.f. = 1/q + 2*q + q^7 - 2*q^9 - q^15 + 4*q^17 - 6*q^25 + q^31 + 8*q^33 + ...

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. A029838, A083365, A131125. A208850, A210063.

a[ n_] := SeriesCoefficient[ 2 q^(1/2) EllipticTheta[ 3, 0, q] / EllipticTheta[ 2, 0, q^2], {q, 0, n}]; (* Michael Somos, Jul 05 2014 *)

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

Expansion of q^(1/2) * eta(q^2)^5 / (eta(q)^2 * eta(q^4) * eta(q^8)^2) in powers of q.
Given g.f. A(x), then B(q) = (A(q^2) / q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - (v - 4) * (u - 4)^2.
Given g.f. A(x), then B(q) = A(q^2) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (u^3 - v) * (v^3 + u) - 3*u*v * (2*(u^2 + v^2) - 11). - Michael Somos, Jul 05 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A208850. - Michael Somos, Jul 05 2014
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) = 2 * A083365(n).
Convolution square is A131125. Convolution inverse is A210063. - Michael Somos, Jul 05 2014

A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

Original entry on oeis.org

1, 1, 0, 1, 1, 0, 1, 2, -1, 0, 1, 3, -1, 0, 0, 1, 4, 0, -2, 1, 0, 1, 5, 2, -5, 3, 0, 0, 1, 6, 5, -8, 3, 2, -1, 0, 1, 7, 9, -10, -1, 9, -4, -1, 0, 1, 8, 14, -10, -10, 20, -7, -4, 2, 0, 1, 9, 20, -7, -24, 31, -2, -15, 5, 1, 0, 1, 10, 27, 0, -42, 36, 20, -40, 9, 8, -2, 0, 1, 11, 35, 12, -62, 28, 65, -75, 3, 27, -8, -1, 0

Offset: 0

Ilya Gutkovskiy, Dec 04 2017

			G.f. of column k: A_k(x) = 1 + k*x + (1/2)*k*(k - 3)*x^2 + (1/6)*k*(k^2 - 9*k + 8)*x^3 + (1/24)*k*(k^3 - 18*k^2 + 59*k - 18)*x^4 + (1/120)*k*(k^4 - 30*k^3 + 215*k^2 - 330*k + 144)*x^5 + ...
Square array begins:
1,  1,  1,  1,   1,   1,  ...
0,  1,  2,  3,   4,   5,  ...
0, -1, -1,  0,   2,   5,  ...
0,  0, -2, -5,  -8, -10,  ...
0,  1,  3,  3,  -1, -10,  ...
0,  0,  2,  9,  20,  31,  ...

G. C. Greubel, Rows n=0..100 of antidiagonals, flattened

Columns k=0..8 give A000007, A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845 (with offset 0).
Main diagonal gives A296043.
Cf. A296068.

Table[Function[k, SeriesCoefficient[Product[((1 + x^(2 i - 1))/(1 + x^(2 i)))^k, {i, 1, n}], {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten
Table[Function[k, SeriesCoefficient[(x^(1/8) EllipticTheta[2, 0, x^(1/2)]/EllipticTheta[2, 0, x])^k, {x, 0, n}]][j - n], {j, 0, 12}, {n, 0, j}] // Flatten

G.f. of column k: Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

G.f. of column k: (x^(1/8)*theta_2(sqrt(x))/theta_2(x))^k, where theta_() is the Jacobi theta function.

A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 1, 2, 0, 0, -1, -4, 0, 0, 0, 6, 0, 0, 1, -8, 0, 0, 0, 12, 0, 0, -1, -18, 0, 0, -1, 24, 0, 0, 2, -32, 0, 0, 1, 44, 0, 0, -2, -58, 0, 0, -1, 76, 0, 0, 2, -100, 0, 0, 1, 128, 0, 0, -3, -164, 0, 0, -1, 210, 0, 0, 4, -264, 0, 0, 2, 332, 0, 0, -5, -416, 0, 0, -2, 516, 0, 0, 5, -640, 0, 0, 2, 790, 0, 0, -6, -968

Offset: 0

Michael Somos, Mar 20 2004, Oct 22 2007

sign

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

eta(q^2) * eta(q^8)^6 = eta(q)^2 * eta(q^4)^2 * eta(q^8) * eta(q^16)^2 + 2 * eta(q^2) * eta(q^4)^2 * eta(q^16)^4 is equivalent to the a(4*n), ..., a(4*n + 3) results.

			G.f. = 1 - 2*x + x^4 + 2*x^5 - x^8 - 4*x^9 + 6*x^13 + x^16 - 8*x^17 + 12*x^21 - ...
G.f. = 1/q - 2*q + q^7 + 2*q^9 - q^15 - 4*q^17 + 6*q^25 + q^31 - 8*q^33 + ...

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. A029838, A029839, A083365, A131124, A187154.

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

{a(n) = if( n<0, 0, polcoeff( prod( k=1, n, (1 - x^k)^[ 0, 2, 1, 2, 2, 2, 1, 2][1 + k%8], 1 + x * O(x^n)), n))}

{a(n) = my(A, A2, m); if( n<0, 0, A = x + O(x^2); m=1; while( m<=n, m*=2; A = subst(A, x, x^2); A = 4*A + 16*A^2 + (1 + 8*A) * sqrt(A + 4*A^2)); polcoeff( sqrt(x / A), n))}

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

Expansion of q^(1/2) * eta(q)^2 * eta(q^4) / (eta(q^2) * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, -1, -2, -2, -2, -1, -2, 0, ...].
Given g.f. A(x), then B(q) = A(q)^2 / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u * (u + 8) * (v + 4) - v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A187154.
G.f.: Product_{k>0} (1 - x^k)^2 / ((1 - x^(4*k - 2)) * (1 - x^(8*k))^2).
a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A029838(n). a(4*n + 1) =  -2 * A083365(n). Convolution square is A131124. Convolution inverse is A187154.

A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 1, -1, 2, -2, 2, -3, 4, -5, 5, -6, 8, -9, 10, -12, 15, -17, 19, -22, 26, -30, 33, -38, 45, -51, 56, -64, 74, -83, 92, -104, 119, -133, 147, -165, 187, -208, 229, -256, 288, -319, 351, -390, 435, -481, 528, -584, 649, -715, 783, -863, 954, -1047, 1145

Offset: 0

Michael Somos, Nov 06 2015

sign

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

			G.f. = 1 - x + x^2 - x^3 + 2*x^4 - 2*x^5 + 2*x^6 - 3*x^7 + 4*x^8 - 5*x^9 + ...
G.f. = 1/q - q^15 + q^31 - q^47 + 2*q^63 - 2*q^79 + 2*q^95 - 3*q^111 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 16th equation.

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. A029838, A036016, A082303, A106507, A214264.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x, x^8] QPochhammer[ -x^7, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 1)) (1 - x^(8 k + 4)) (1 + x^(8 k + 7)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 1, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 1][k%16 + 1]), n))};

Expansion of f(x^4, x^12) / f(x, x^7) where f(, ) is Ramanujan's general theta function.
Euler transform of period 16 sequence [ -1, 1, 0, 1, 0, 0, -1, 0, -1, 0, 0, 1, 0, 1, -1, 0, ...].
G.f.: 1 / (Product_{k>=0} (1 + x^(8*k + 1)) * (1 - x^(8*k + 4)) * (1 + x^(8*k + 7))).
G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x + x^7 + x^10 + x^22 + ...). [Ramanujan]
G.f.: 1 - x * (1 - x) / (1 - x^2) + x^4 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
a(n) = (-1)^n * A036016(n) = A029838(2*n) = A082303(2*n).
Convolution product of A106507 and A214264.

A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, 0, -1, 1, -1, 1, -1, 2, -2, 2, -3, 4, -4, 4, -6, 7, -7, 8, -10, 12, -13, 14, -17, 21, -22, 24, -29, 33, -36, 40, -46, 53, -58, 63, -73, 83, -90, 99, -113, 127, -138, 152, -171, 191, -209, 228, -255, 285, -309, 338, -377, 416, -453, 495, -547, 603, -656

Offset: 0

Michael Somos, Nov 08 2015

sign

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

			G.f. = 1 - x^3 + x^4 - x^5 + x^6 - x^7 + 2*x^8 - 2*x^9 + 2*x^10 - 3*x^11 + ...
G.f. = q^7 - q^55 + q^71 - q^87 + q^103 - q^119 + 2*q^135 - 2*q^151 + ...

Srinivasa Ramanujan, The Lost Notebook and Other Unpublished Papers, Narosa Publishing House, New Delhi, 1988, p. 41, 15th equation.

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. A029838, A036015, A082303, A106507, A214263.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x^4, x^4] / (QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8]), {x, 0, n}];
a[ n_] := SeriesCoefficient[ 1 / Product[ (1 + x^(8 k + 3)) (1 - x^(8 k + 4)) (1 + x^(8 k + 5)), {k, 0, Ceiling[ n/8]}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0, 0}[[Mod[k, 16, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, -1, 1, -1, 1, 0, 0][k%16 + 1]), n))};

Expansion of f(x^4, x^12) / f(x^3, x^5) where f(, ) is Ramanujan's general theta function.
Euler transform of period 16 sequence [ 0, 0, -1, 1, -1, 1, 0, 0, 0, 1, -1, 1, -1, 0, 0, 0, ...].
G.f.: (1 + x^4 + x^12 + x^24 + x^40 + ...) / (1 + x^3 + x^5 + x^14 + x^18 + ...). [Ramanujan]
G.f.: 1 - x^3 * (1 - x) / (1 - x^2) + x^8 * (1 - x) * (1 - x^3) / ((1 - x^2) * (1 - x^4)) - ... [Ramanujan]
a(n) = (-1)^n * A036015(n) = A029838(2*n + 1) = - A082303(2*n + 1).
Convolution product of A106507 and A214263.

A258939 Expansion of f(-x^3, -x^5) * f(x^3, x^13) / (f(-x, -x^2) * f(-x^8, -x^16)) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 2, 3, 5, 6, 9, 12, 17, 22, 30, 38, 51, 64, 83, 104, 133, 165, 208, 256, 319, 390, 481, 584, 715, 863, 1047, 1258, 1517, 1812, 2172, 2584, 3080, 3648, 4327, 5104, 6028, 7084, 8330, 9756, 11430, 13340, 15574, 18122, 21086, 24464, 28378, 32832, 37977, 43823

Offset: 0

Michael Somos, Nov 07 2015

nonn

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

			G.f. = 1 + x + 2*x^2 + 3*x^3 + 5*x^4 + 6*x^5 + 9*x^6 + 12*x^7 + 17*x^8 + ...
G.f. = q^15 + q^47 + 2*q^79 + 3*q^111 + 5*q^143 + 6*q^175 + 9*q^207 + ...

Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A029838.

a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^-{ 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0}[[Mod[k, 32, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^-[ 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1][k%32 + 1]), n))};

Euler transform of period 32 sequence [ 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 1, 1, 0, ...].
- a(n) = A029838(4*n + 2).
a(n) ~ sqrt(2*(1+sqrt(2))) * exp(Pi*sqrt(n/2)) / (16*n^(3/4)). - Vaclav Kotesovec, Nov 07 2015

A279255 Expansion of chi(x) * chi(-x^3) * chi(-x^8) * chi(-x^24) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, 1, 0, 0, 0, -1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, -1, -1, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, -1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 0, -3, 0, 0, 0, -1, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, -1, -5, 0, 0, 0, -2, 0, 0, 0, 5, 0

Offset: 0

Michael Somos, Dec 08 2016

sign

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

			G.f. = 1 + x + x^5 - x^9 + x^12 + x^17 - x^24 - x^25 - x^29 + 2*x^33 + ...
G.f. = q^-3 + q^-1 + q^7 - q^15 + q^21 + q^31 - q^45 - q^47 - q^55 + ...

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. A029838.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ x^3, x^6] QPochhammer[ x^8, x^16] QPochhammer[ x^24, x^48], {x, 0, n}];

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^8 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^16 + A) * eta(x^48 + A)), n))};

Euler transform of period 48 sequence [ 1, -1, 0, 0, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, -2, 1, -1, 0, 0, 1, -1, 1, 0, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 0, 0, -1, 1, 0, ...].
a(4*n + 2) = a(4*n + 3) = a(6*n + 2) = a(6*n + 4) = 0.
a(4*n + 1) = a(12*n) = A029838(n).

A283023 Expansion of f(-x, -x^5)^2 / (f(x^2, x^10) * f(x^6, x^18)) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -2, 0, 2, 0, -4, 1, 6, 0, -8, 0, 12, -1, -18, 0, 24, 0, -32, 0, 44, 0, -58, 0, 76, 1, -100, 0, 128, 0, -164, 0, 210, 0, -264, 0, 332, -1, -416, 0, 516, 0, -640, -1, 790, 0, -968, 0, 1184, 2, -1444, 0, 1752, 0, -2120, 1, 2560, 0, -3078, 0, 3692, -2, -4420, 0

Offset: 0

Michael Somos, Feb 26 2017

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

			G.f. = 1 - 2*x + 2*x^3 - 4*x^5 + x^6 + 6*x^7 - 8*x^9 + 12*x^11 + ...
G.f. = q^-3 - 2*q + 2*q^9 - 4*q^17 + q^21 + 6*q^25 - 8*q^33 + 12*q^41 + ...

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. A029838, A081055, A081056, A083365, A134178, A258741, A258939, A259774.

a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^2 QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / QPochhammer[ -x^2, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^3, x^6]^2 QPochhammer[ x^12, x^24] / EllipticTheta[ 4, 0, x^4], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(1/2) x^(3/4) EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^3] / (EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}] // Simplify;

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) / (eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^4 + A)^2 * eta(x^12 + A) * eta(x^24 + A)), n))};

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^2*eta(q^6)^4*eta(q^8)/(eta(q^2)*eta(q^3)^2*eta(q^4)^2*eta(q^12)*eta(q^24)))} \\ Altug Alkan, Mar 21 2018

Expansion of chi(-x)^2 * chi(x^3)^2 * chi(-x^12) / chi(x^2) in powers of x where chi() is a Ramanujan theta function.
Expansion of phi(-x) * chi(x^3)^2 * chi(-x^12) / phi(-x^4) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of phi(-x) * phi(x^3) / (phi(-x^4) * psi(-x^6)) ih powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(3/4) * eta(q)^2 * eta(q^6)^4 * eta(q^8) / (eta(q^2) * eta(q^3)^2 * eta(q^4)^2 * eta(q^12) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [-2, -1, 0, 1, -2, -3, -2, 0, 0, -1, -2, 0, -2, -1, 0, 0, -2, -3, -2, 1, 0, -1, -2, 0, ...].
a(n) = A134178(2*n).  a(6*n + 2) = A(6*n + 4) = 0.
a(2*n + 1) = -2 * A083365(n). a(4*n + 1) = -2 * A081055(n). a(4*n + 3) = 2 * A081056(n).
a(6*n) = A029838(n). a(12*n) = A258741(n). a(12*n + 6) = A259774(n). a(24*n + 12) = - A258939(n).

A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.

Original entry on oeis.org

1, 1, -1, 1, 1, 0, 0, -2, 5, 0, -2, 0, 3, 5, -11, 5, 6, 9, -17, -2, 23, -3, -11, -25, 62, -11, -27, -27, 76, 20, -104, 10, 77, 101, -243, 58, 118, 147, -353, -25, 378, 48, -372, -298, 892, -165, -444, -621, 1524, -128, -1055, -559, 1869, 575, -2682, 84, 2054, 1979, -5325, 844, 2947

Offset: 0

Ilya Gutkovskiy, Dec 03 2017

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Cf. A029838, A029839, A029840, A029841, A029842, A029843, A029844, A029845, A295831, A295832, A296046.

nmax = 60; CoefficientList[Series[Product[((1 + x^(2 k - 1))/(1 + x^(2 k)))^k, {k, 1, nmax}], {x, 0, nmax}], x]

G.f.: Product_{k>=1} ((1 + x^(2*k-1))/(1 + x^(2*k)))^k.

cp's OEIS Frontend

A134178 Expansion of chi(x) * chi(-x^2)^2 * chi(-x^3) * chi(-x^4) * chi(x^6)^2 * chi(-x^12) in powers of x where chi() is a Ramanujan theta function.

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A208589 Expansion of phi(x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2*j-1))/(1 + x^(2*j)))^k.

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A093085 Expansion of phi(-x) / psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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A258741 Expansion of f(x^3, x^5) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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A259774 Expansion of f(x, x^7) / f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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A258939 Expansion of f(-x^3, -x^5) * f(x^3, x^13) / (f(-x, -x^2) * f(-x^8, -x^16)) in powers of x where f(, ) is Ramanujan's general theta function.

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A296067 Square array A(n,k), n>=0, k>=0, read by antidiagonals, where column k is the expansion of Product_{j>=1} ((1 + x^(2j-1))/(1 + x^(2j)))^k.

A296047 Expansion of Product_{k>=1} ((1 + x^(2k-1))/(1 + x^(2k)))^k.