A112608 -id:A112608 - cp's OEIS frontend

A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

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

Offset: 1

Michael Somos, Jan 29 2008

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. = q - q^2 + q^3 - q^4 - q^6 + 2*q^7 - q^8 + q^9 - q^12 + 2*q^13 + ...

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. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.

Cf. A035178, A093829, A113447 are same up to sign.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)

{a(n) = if( n<1, 0, -(-1)^n * sumdiv(n, d, kronecker(-12, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^12 + A))) / 3, n))}; /* Michael Somos, May 06 2015 */

Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless e=0, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( Sum_{k>0} x^(6*k-5) / ( 1 + x^(6*k-5) ) - x^(6*k-1) / ( 1 + x^(6*k-1) )).
a(n) = -(-1)^n * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). a(8*n + 7) = 2 * A112609(n).
a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n).
a(24*n + 1) = A131961(n). a(24*n + 7) = 2 * A131962(n). a(24*n + 13) = 2 * A131963(n). a(24*n + 19) = 2 * A131964(n).

A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 1, 0, 0, 3, -4, 0, 2, -2, 0, 2, 0, 0, 1, -4, 0, 0, 0, 0, 3, 0, 0, 2, -4, 0, 4, -2, 0, 2, 0, 0, 0, -8, 0, 1, 0, 0, 2, 0, 0, 2, 0, 0, 3, 0, 0, 0, 0, 0, 2, -4, 0, 2, -6, 0, 2, 0, 0, 4, -4, 0, 0, -4, 0, 1, 0, 0, 4, 0, 0, 2, 0, 0, 2, 0, 0, 1, -4, 0, 0, -4

Offset: 0

Michael Somos, Aug 04 2015

sign

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

			G.f. = 1 - 2*x + x^3 + 3*x^6 - 4*x^7 + 2*x^9 - 2*x^10 + 2*x^12 + x^15 + ...
G.f. = q - 2*q^9 + q^25 + 3*q^49 - 4*q^57 + 2*q^73 - 2*q^81 + 2*q^97 + ...

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. A112608, A131961.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 3, 0, x^6] QPochhammer[ -x^3, x^3], {x, 0, n}];

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

q='q+O('q^99); Vec(eta(q)^2*eta(q^12)^5/(eta(q^2)*eta(q^3)*eta(q^6)*eta(q^24)^2)) \\ Altug Alkan, Aug 01 2018

Expansion of q^(-1/8) * eta(q)^2 * eta(q^12)^5 / (eta(q^2) * eta(q^3) * eta(q^6) * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -4, -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -2, ...].
a(3*n) = A131961(n). a(3*n + 1) = -2 * A112608(n). a(3*n + 2) = 0.

A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 28 2015

sign

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

			G.f. = 1 - 2*x + 2*x^4 + x^6 - 2*x^7 - 2*x^9 + 2*x^10 - 2*x^15 + 2*x^16 + ...
G.f. = q^3 - 2*q^7 + 2*q^19 + q^27 - 2*q^31 - 2*q^39 + 2*q^43 - 2*q^63 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A112605, A112608, A112609, A138270, A262780.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 3, KroneckerSymbol[ -3, #] &]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^3] / (2 x^(3/4)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 3))]; (* Michael Somos, Oct 01 2015 *)

{a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 3, d, kronecker(-3, d)))};

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 3); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))}; /* Michael Somos, Oct 01 2015 */

Expansion of q^(-3/4) * eta(q)^2 * eta(q^12)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [-2, -1, -2, -1, -2, 0, -2, -1, -2, -1, -2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A112605(n). -2 * a(n) = A138270(2*n + 1).
a(2*n) = A112608(n). a(2*n + 1) = -2 * A112609(n). a(3*n + 2) = 0.
a(n) = A262780(2*n + 1). - Michael Somos, Oct 01 2015

A262774 Expansion of psi(x^2) * phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 01 2015

sign

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

			G.f. = 1 + x^2 - 2*x^3 - 2*x^5 + x^6 - 2*x^9 + 3*x^12 + 2*x^14 - 2*x^15 + ...
G.f. = q + q^9 - 2*q^13 - 2*q^21 + q^25 - 2*q^37 + 3*q^49 + 2*q^57 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A112604, A112604, A112606, A112607, A112608, A112609, A131961, A131963.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 4 n + 1, KroneckerSymbol[ -3, #]&]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 4, 0, x^3] / (2 x^(1/4)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^3]^2 QPochhammer[ x^4]^2 / ( QPochhammer[ x^2]  QPochhammer[ x^6]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 5, Mod[#, 2], Mod[#, 6] == 5, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 6, #]^#2] & @@@ FactorInteger @ (4 n + 1))];

{a(n) = if( n<0, 0, (-1)^n * sumdiv(4*n + 1, d, kronecker( -3, d)))};

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor(4*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, p%2, p%6 == 1, (e+1) * if( p%24 == 1 || p%24 == 19, 1, (-1)^e), 1-e%2 )))};

Expansion of q^(-1/4) * eta(q^3)^2 * eta(q^4)^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, -2, -1, 0, 0, 0, -1, -2, 1, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 192^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(n) = (-1)^n * A112604(n). a(2*n) = A112606(n). a(2*n + 1) = -2 * A112607(n-1). a(3*n + 1) = 0.
a(6*n) = A131961(n).  a(6*n + 2) = A112608(n). a(6*n + 3) = -2 * A131963(n). a(6*n + 5) = -2 * A112609(n).

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A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

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A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

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A262726 Expansion of phi(-x) * psi(x^6) in powers of x where phi(), psi() are Ramanujan theta functions.

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A262774 Expansion of psi(x^2) * phi(-x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

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