A128582 -id:A128582 - cp's OEIS frontend

A128580 Expansion of phi(x^3) * psi(x^4) - x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

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

Offset: 0

Michael Somos, Mar 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^4 - 2*x^5 + 2*x^7 - 2*x^10 + 3*x^12 + ...
G.f. = q - q^3 - 2*q^5 + 2*q^7 + q^9 - 2*q^11 + 2*q^15 - 2*q^21 + 3*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. A113780, A115660, A128581, A128582, A259668.

a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 12 2015 *)

{a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker(-12, d) * kronecker(2, n/d)))};

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

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1)* (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, -2, 0, 0, -1, -1, -1, -1, 0, -2, -1, -2, -1, -2, 0, -1, -1, -1, -1, 0, 0, -2, -1, -2, ...].
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11)= 0.
G.f.: Product_{k>0} (1 - x^(8*k)) * (1 - x^(12*k))^2 / ((1 + x^k) * (1 + x^(2*k))^2 * (1 - x^(3*k)) * (1 + x^(12*k))).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (x^k - x^(3*k)) / (1 + x^(4*k)) * Kronecker(-12, k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) * Kronecker(2, k).
a(n) = A128581(2*n + 1) = A115660(2*n + 1). a(3*n + 2) = -2 * A128582(n). a(12*n) = A113780(n).
a(2*n) = A259668(n). a(3*n + 1) = - A128580(n). - Michael Somos, Jul 12 2015

A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 13 2007

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^4 + 2*x^5 + 2*x^7 + 2*x^10 + 3*x^12 + x^13 + 2*x^14 + ...
G.f. = q + q^3 + 2*q^5 + 2*q^7 + q^9 + 2*q^11 + 2*q^15 + 2*q^21 + 3*q^25 + q^27 + ...

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).

Antti Karttunen, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000377, A128580, A128582, A113780, A190615, A257920, A259895, A260308, A261115, A261118, A261119, A261122.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ x^3] QPochhammer[ -x, x] QPochhammer[ x^6, -x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)

{a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -6, d)))};

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

Expansion of f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A190611.
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).
a(n) = A000377(2*n + 1). a(3*n + 2) = 2 * A128582(n). a(12*n) = A113780(n).
a(n) = (-1)^n * A190615(n) = (-1)^floor( (n+1) / 2) * A128580(n). - Michael Somos, Nov 11 2015
a(2*n) = A261118(n). a(2*n + 1) = A261119(n). a(3*n) = A261115(n). - Michael Somos, Nov 11 2015
a(4*n) = A260308(n). a(4*n + 1) = A257920(n). a(4*n + 2) = 2 * A259895(n). - Michael Somos, Nov 11 2015
a(n) = - A261122(4*n + 2). - Michael Somos, Nov 11 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 1, 1, 0, 3, 1, 1, 1, 2, 2, 1, 2, 0, 1, 2, 1, 0, 1, 2, 3, 0, 1, 1, 1, 3, 2, 1, 1, 1, 1, 2, 0, 2, 1, 2, 0, 1, 0, 1, 4, 1, 2, 0, 1, 2, 1, 2, 1, 1, 3, 0, 1, 2, 3, 1, 0, 1, 0, 0, 2, 2, 1, 1, 2, 2, 1, 1, 2, 0, 1, 2, 0, 1, 1, 6, 1, 1, 1, 0, 2, 1

Offset: 0

Michael Somos, Mar 11 2007

nonn

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

			G.f. = 1 + x + x^2 + 2*x^3 + x^4 + 2*x^5 + x^6 + x^7 + x^8 + 3*x^10 + ...
G.f. = q^5 + q^29 + q^53 + 2*q^77 + q^101 + 2*q^125 + q^149 + q^173 + ...

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. A128580, A128582, A190615, A229723, A259895, A260118.

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

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

Expansion of q^(-5/24) * eta(q^2) * eta(q^4) * eta(q^6)^2 / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, 0, 1, -1, 1, -2, 1, -1, 1, 0, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 6^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A229723.
a(n) = A128582(4*n) = A259895(3*n) = A260118(4*n). 2 * a(n) = A190615(12*n + 2). - Michael Somos, Nov 15 2015
-2 * a(n) = A128580(12*n + 2). - Michael Somos, Dec 22 2016

A128591 Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Mar 11 2007

nonn

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

			G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + x^10 + 3*x^11 + ...
G.f. = q^11 + 2*q^35 + q^59 + q^83 + q^107 + q^131 + 2*q^155 + q^179 + 2*q^203 + ...

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. A128580, A128582, A190615, A257920, A257921,A260118.

a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 15 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)  / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};

Expansion of chi(x) * psi(x) * psi(-x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Nov 15 2015
Expansion of  q^(-11/24) * eta(q^2)^4 * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 1, -1, 2, -2, 2, -1, 1, -2, 2, -2, ...].
a(n) = A128582(4*n + 1).
2 * a(n) = A257920(3*n + 1). - a(n) = A260118(4*n + 1). 2 * a(n) = A257921(6*n + 2). -2 * a(n) = A128580(12*n + 5) = A190615(12*n + 5). - Michael Somos, Nov 15 2015

A263571 Expansion of f(x^2, x^2) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 2, 2, 0, 1, 0, 2, 3, 2, 2, 0, 0, 2, 0, 0, 3, 0, 4, 2, 0, 1, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 3, 2, 2, 0, 2, 0, 2, 3, 2, 2, 0, 0, 0, 0, 0, 4, 0, 2, 4, 0, 2, 0, 2, 1, 0, 6, 0, 0, 0, 0, 0, 2, 3, 2, 2, 0, 0, 0, 2, 4, 4, 2, 0, 0, 2, 0, 0, 2, 0, 0, 4, 0, 1, 0

Offset: 0

Michael Somos, Oct 21 2015

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^7 + 3*x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^10 + q^16 + 2*q^22 + 3*q^25 + 2*q^28 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A113780, A115660, A128581, A128582, A192013, A260089, A261115, A263548, A263577.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, m/d], {d, Divisors[ m]}]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}];

{a(n) = my(m); if( n<0, 0, m = 3*n + 1; sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};

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

Expansion of chi(x) * phi(x^2) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q^3) * eta(q^4)^4 * eta(q^12) / (eta(q) * eta(q^6) * eta(q^8)^2) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263577.
a(n) = A115660(3*n + 1) = A192013(3*n + 1) = A128581(6*n + 2).
a(2*n) = A261115(n). a(2*n + 1) = A263548(n). a(4*n + 1) = a(n). a(4*n + 3) = 2 * A128582(n).
a(8*n + 4) = a(8*n + 6) = 0. a(8*n) = A113780(n). a(8*n + 2) = 2 * A260089(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A260118 Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jul 16 2015

sign

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

			G.f. = 1 - x + x^4 - 2*x^5 + x^8 - x^9 + 2*x^12 - x^13 + x^16 - x^17 + ...
G.f. = q^5 - q^11 + q^29 - 2*q^35 + q^53 - q^59 + 2*q^77 - q^83 + q^101 + ...

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. A128538, A128582, A128591, A134177, A190615.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 2, 0, x^(3/2)]/(4 x^(7/8) QPochhammer[ -x]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, (-1)^n DivisorSum[ m, {-1, 0, 0, 0, 1, 0}[[Mod[#, 6, 1]]] {1, 0, 0, 0, 0, 0, 1, 0}[[Mod[m/#, 8, 1]]] &]]];

{a(n) = my(m); if( n<0, 0, m=6*n + 5; (-1)^n * sumdiv(m, d, ((d%6==5) - (d%6==1)) * ((m/d%8==1) + (m/d%8==7))))};

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

Expansion of psi(-x^2)^2 * psi(x^3) / f(x) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of psi(-x) * psi(x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q) * eta(q^6)^2 * eta(q^8)^2 / (eta(q^2) * eta(q^3) * eta(q^4)) in powers of q.
Euler transform of period 24 sequence [ -1, 0, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, 1, 0, 0, -1, -2, ...].
a(n) = (-1)^n * A128582(n). 2 * a(n) = - A134177(3*n + 2) = A190615(3*n + 2).
a(4*n) = A128583(n). a(4*n + 1) = - A128591(n). a(4*n + 2) = a(4*n + 3) = 0.

A258587 Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

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 - 2*x + x^2 - 2*x^3 + 2*x^4 + 2*x^6 - 2*x^9 + x^10 - 4*x^11 + 2*x^14 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^11 + 2*q^14 + 2*q^20 - 2*q^29 + q^32 - 4*q^35 + ...

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. A128581, A128582, A190611, A263548, A263571.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];

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

Expansion of phi(-x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ -2, 0, -2, -2, -2, -1, -2, -1, -2, 0, -2, -2, -2, 0, -2, -1, -2, -1, -2, -2, -2, 0, -2, -2, ...].
a(n) = (-1)^n * A263548(n) = A128581(3*n + 2) = A190611(3*n + 2).
a(2*n) = A263571(n). a(2*n + 1) = -2 * A128582(n).

A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2(-1)^e if e>0, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 22 2015

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

			G.f. = x - x^2 + 2*x^3 + x^4 - 2*x^5 - 2*x^6 + 2*x^7 - x^8 - 2*x^9 + ...

G. C. Greubel, Table of n, a(n) for n = 1..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A115660, A128580, A128582, A261115, A263548, A263571.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {-1, 1, -2}[[#]] (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])];

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-1)^e, p==3, -2 * (-1)^e, p%24>12, 1-e%2, (e+1) * kronecker(p, 12)^e )))};

a(2*n) = - a(n). a(3*n) = 2 * A115660(n). a(3*n + 1) = A263571(n+1). a(3*n + 2) = - A263548(n).
a(6*n + 1) = A261115(n). a(6*n + 3) = 2 * A128580(n). a(6*n + 5) = -2 * A128582(n).
Sum_{k=1..n} abs(a(k)) ~ (2/3)*sqrt(2/3)*Pi*n. - Amiram Eldar, Jan 29 2024

A279947 Expansion of f(x^2, x^2) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Dec 23 2016

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^7 + 3*x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 - 2*q^10 - q^16 - 2*q^22 + 3*q^25 - 2*q^28 + ...

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. A113780, A128581, A128582, A128583, A128591, A190611, A260089, A261115, A261122, A263548, A263571.

a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 2, #] KroneckerSymbol[ -3, m/#] &]]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ x^6] QPochhammer[ x, x^6] QPochhammer[ x^5, x^6], {x, 0, n}];

{a(n) = my(m); if( n<0, 0, m = 3*n + 1; (-1)^n * sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor(3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, (-1)^e, p%24==1 || p%24==7, e+1, p%24==5 || p%24==11, (e+1)*(-1)^e, !(e%2))))};

Expansion of chi(-x) * phi(x^2) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 2, 0, -3, -1, 1, -1, -1, 0, 2, -1, -4, -1, 2, 0, -1, -1, 1, -1, -3, 0, 2, -1, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
a(n) = (-1)^n * A263571(n) = A128581(3*n + 1) = - A190611(3*n + 1) = - A261122(6*n + 2).
a(2*n) = A261115(n).
a(2*n + 1) = - A263548(n).
a(8*n + 4) = a(8*n + 6) = 0.
a(4*n + 1) = -a(n).
a(4*n + 3) = -2 * A128582(n).
a(8*n) = A113780(n).
a(8*n + 2) = 2 * A260089(n).
a(16*n + 3) = -2 * A128583(n).
a(16*n + 7) = -2 * A128591(n).

A258940 Expansion of f(-x^8) * f(-x^12) * f(-x^24) * f(-x^2, -x^6)^2 / (f(-x^2) * f(-x^3, -x^5) * f(-x^3, -x^21)) in powers of x where f() is a Ramanujan theta function and f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 07 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 + x^4 - x^5 + x^6 + x^7 + x^9 + 2*x^12 - x^14 + ...
G.f. = q - q^5 + 2*q^7 + q^9 - q^11 + q^13 + q^15 + q^19 + 2*q^25 - q^29 + ...

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

Cf. A128582.

a[ n_] := SeriesCoefficient[ Product[(1 - x^k)^-{ 0, -1, 2, 1, 1, -1, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, -1, 1, 1, 2, -1, 0, -2}[[Mod[k, 24, 1]]], {k, n}], {x, 0, n}];

{a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[ 2, 0, 1, -2, -1, -1, 1, 0, 1, 0, 1, -1, 0, -1, 1, 0, 1, 0, 1, -1, -1, -2, 1, 0][k%24 + 1]), n))};

Euler transform of period 24 sequence [ 0, -1, 2, 1, 1, -1, 0, -1, 0, -1, 1, 0, 1, -1, 0, -1, 0, -1, 1, 1, 2, -1, 0, -2, ...].
a(3*n + 2) = - A128582(n).
a(12*n + 8) = a(12*n + 11) = 0.

cp's OEIS Frontend

A128580 Expansion of phi(x^3) * psi(x^4) - x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

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A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

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A128583 Expansion of chi(x) * psi(x^2) * phi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A128591 Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

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A263571 Expansion of f(x^2, x^2) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

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A260118 Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.

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A258587 Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

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A263649 a(n) is multiplicative with a(2^e) = (-1)^e, a(3^e) = -2(-1)^e if e>0, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) (-1)^e if p == 5, 11 (mod 24), a(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).