A260089 -id:A260089 - cp's OEIS frontend

A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() 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

Offset: 0

Michael Somos, May 14 2011

sign

Number 63 of the 74 eta-quotients listed in Table I of Martin (1996).

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 + ...
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
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 44.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113700, A128580, A128583, A128591, A129402, A134177, A257920, A259895, A259896, A260089, A260110, A260118, A260308.

a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ -x^3] / (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^12]), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)

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

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

{a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, (-1)^e, p%24 < 12, (e+1) * if( p%24 < 6, 1, (-1)^e), (1 + (-1)^e) / 2 )))};

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.
Expansion of q^(-1/2) * eta(q) * eta(q^4)^4 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3) * eta(q^8) * eta(q^12)^3) in powers of q.
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, 5 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 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, -2, -1, -1, -1, -1, 0, 2, -1, -2, -1, 2, 0, -1, -1, -1, -1, -2, 0, 2, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker( 6, k) * q^k / (1 + q^(2*k)) = Sum_{k>=0} a(k) * q^(2*k + 1).
G.f.: Product_{k>0}  (1 + (-x)^k) * (1 - (-x^2)^k) * (1 - (-x^3)^k) * (1 + (-x^6)^k).
a(n) = (-1)^n * A129402(n). a(3*n + 1) = -a(n). a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
a(12*n) = A113700(n). a(12*n + 2) = 2 * A128583(n). a(12*n + 5) = -2 * A128591(n). - Michael Somos, Jun 09 2015
a(n) = (-1)^floor(n/2) * A128580(n) = (-1)^(n + floor(n/2)) * A134177(n). - Michael Somos, Jul 29 2015
a(3*n) = A260110(n). a(3*n + 2) = 2 * A260118(n). - Michael Somos, Jul 29 2015
a(4*n) = A260308(n). a(4*n + 1) = - A257920(n). a(4*n + 2) = 2 * A259895(n). a(4*n + 3) = -2 * A259896(n). - Michael Somos, Jul 29 2015
a(12*n + 3) = -2 * A260089(n). - Michael Somos, Jul 29 2015

A261115 Expansion of f(x, x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 08 2015

nonn

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

			G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 3*x^8 + 4*x^9 + 2*x^12 + 2*x^13 + 2*x^16 + ...
G.f. = q + 2*q^7 + 3*q^25 + 2*q^31 + 3*q^49 + 4*q^55 + 2*q^73 + 2*q^79 + ...

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, A128580, A128581, A129402, A192013, A260110.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];

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

Expansion of q^(-1/6) * eta(q^2)^5 * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -2, ...].
a(n) = (-1)^n * A260110(n) = A128580(3*n) = A129402(3*n) = A115660(6*n + 1) = A128581(6*n + 1) = A192013(6*n + 1).
a(4*n) = A113780(n). a(4*n + 1) = 2 * A260089(n). a(4*n + 2) = a(4*n + 3) = 0.

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

A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

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 - 2*x + 3*x^4 - 2*x^5 + 3*x^8 - 4*x^9 + 2*x^12 - 2*x^13 + 2*x^16 + ...
G.f. = q - 2*q^7 + 3*q^25 - 2*q^31 + 3*q^49 - 4*q^55 + 2*q^73 - 2*q^79 + ...

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. A134177, A190615, A229723, A260089.

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

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

Expansion of q^(-1/6) * eta(q)^2 * eta(q^8) * eta(q^12)^2 / (eta(q^2) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -2, ...].
a(n) = A134177(3*n) = A190615(3*n) = A229723(6*n + 1). a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A113780(n). a(4*n + 1) = -2 * A260089(n).

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).

cp's OEIS Frontend

A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.

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A261115 Expansion of f(x, x) * f(x^4, x^8) 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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A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

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A279947 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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