A123530 -id:A123530 - cp's OEIS frontend

A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

1, -3, 0, 6, -3, 0, 0, -6, 0, 6, 0, 0, 6, -6, 0, 0, -3, 0, 0, -6, 0, 12, 0, 0, 0, -3, 0, 6, -6, 0, 0, -6, 0, 0, 0, 0, 6, -6, 0, 12, 0, 0, 0, -6, 0, 0, 0, 0, 6, -9, 0, 0, -6, 0, 0, 0, 0, 12, 0, 0, 0, -6, 0, 12, -3, 0, 0, -6, 0, 0, 0, 0, 0, -6, 0, 6, -6, 0, 0, -6, 0, 6, 0, 0, 12, 0, 0, 0, 0, 0, 0, -12, 0, 12, 0, 0, 0, -6, 0, 0

Offset: 0

N. J. A. Sloane

Unsigned sequence is expansion of theta series of hexagonal net with respect to a node.
Cubic AGM theta functions: a(q) (see A004016), b(q) (this: A005928), c(q) (A005882).
Denoted by a_3(n) in Kassel and Reutenauer 2015. - Michael Somos, Jun 04 2015

			G.f. = 1 - 3*q + 6*q^3 - 3*q^4 - 6*q^7 + 6*q^9 + 6*q^12 - 6*q^13 - 3*q^16 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 79, Eq. (32.34).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
J. M. Borwein and P. B. Borwein, A cubic counterpart of Jacobi's identity and the AGM, Trans. Amer. Math. Soc., 323 (1991), no. 2, 691-701. MR1010408 (91e:33012) see page 695.
J. M. Borwein, P. B. Borwein and F. Garvan, Some Cubic Modular Identities of Ramanujan, Trans. Amer. Math. Soc. 343 (1994), 35-47.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [A later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication below.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.

Cf. A005882, A033762, A097195, A113062, A123477, A123530, A246838, A253243.

A := Basis( ModularForms( Gamma1(9), 1), 100); A[1] - 3*A[2] + 6*A[4]; // Michael Somos, Jan 31 2015

a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 / QPochhammer[ q^3], {q, 0, n}]; (* Michael Somos, May 24 2013 *)
a[ n_] := If[ n < 1, Boole[ n==0], -3 Sum[{1, -1, -3, 1, -1, 3, 1, -1, 0}[[ Mod[ d, 9, 1]]], {d, Divisors @ n}]]; (* Michael Somos, Sep 23 2013 *)

{a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==3, -2, if( p%6==1, e+1, !(e%2)))))}; \\ Michael Somos, May 20 2005

{a(n) = my(A = x * O(x^n)); polcoeff( eta(x + A)^3 / eta(x^3 + A), n)}; \\ Michael Somos, May 20 2005

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, -3, 3, 9, -3, 3, -9, -3, 3] [d%9 + 1]))}; \\ Michael Somos, Dec 25 2007

N=66; x='x+O('x^N); gf=exp(sum(n=1,N,(sigma(n)-sigma(3*n))*x^n/n));
Vec(gf) \\ Joerg Arndt, Jul 30 2011

lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3/eta(q^3))} \\ Altug Alkan, Mar 20 2018

a(n) is the coefficient of q^n in b(q)=eta(q)^3/eta(q^3) = (3/2)*a(q^3)-a(q)/2 where a(q)=theta(Hexagonal). - Kok Seng Chua (chuaks(AT)ihpc.nus.edu.sg), May 07 2002
From Michael Somos, May 20 2005: (Start)
Euler transform of period 3 sequence [ -3, -3, -2, ...].
a(n) = -3 * b(n) except for a(0) = 1, where b()=A123477() is multiplicative with b(p^e) = -2 if p = 3 and e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e)/2 if p == 2, 5 (mod 6).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^3 - 2*u*w^2 + u^2*w.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1^2*u6 - 2*u1*u2*u6 + 4*u2^2*u6 - 3*u2*u3^2.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = u1*u2*u3 + u1^2*u3 - 3*u1*u6^2 + u2^2*u3. (End)
a(3*n + 2) = 0. a(3*n + 1) = -A005882(n), a(3*n) = A004016(n). - Michael Somos, Jul 15 2005
a(n) = -3 * A123477(n) unless n=0. |a(n)| = A113062(n).
Moebius transform is period 9 sequence [-3, 3, 9, -3, 3, -9, -3, 3, 0, ...]. - Michael Somos, Dec 25 2007
Expansion of b(q) = a(q^3) - c(q^3) in powers of q where a(), b(), c() are cubic AGM theta functions. - Michael Somos, Dec 25 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 3^(3/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033687.
G.f.: exp( Sum_{n>=1} (sigma(n)-sigma(3*n))*x^n/n ). - Joerg Arndt, Jul 30 2011
a(n) = (-1)^(mod(n, 3) = 1) * A113062(n). - Michael Somos, Sep 05 2014
a(2*n + 1) = -3 * A123530(n). a(4*n) = a(n). a(4*n + 1) = -3 * A253243(n). a(4*n + 2) = 0. a(4*n + 3) = 6 * A246838(n). a(6*n + 1) = -3 * A097195(n). a(6*n + 3) = 6 * A033762(n). - Michael Somos, Jun 04 2015
G.f.: 1 + Sum_{k>0} -3 * x^k / (1 + x^k + x^(2*k)) + 9 * x^(3*k) / (1 + x^(3*k) + x^(6*k)). - Michael Somos, Jun 04 2015
a(0) = 1, a(n) = -(3/n)*Sum_{k=1..n} A078708(k)*a(n-k) for n > 0. - Seiichi Manyama, Apr 29 2017

Edited by M. F. Hasler, May 07 2018

A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 0, -6, -3, 0, 0, 6, 0, -6, 0, 0, 6, 6, 0, 0, -3, 0, 0, 6, 0, -12, 0, 0, 0, 3, 0, -6, -6, 0, 0, 6, 0, 0, 0, 0, 6, 6, 0, -12, 0, 0, 0, 6, 0, 0, 0, 0, 6, 9, 0, 0, -6, 0, 0, 0, 0, -12, 0, 0, 0, 6, 0, -12, -3, 0, 0, 6, 0, 0, 0, 0, 0, 6, 0, -6, -6, 0, 0, 6, 0

Offset: 0

Michael Somos, Sep 22 2013

sign

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

Zagier (2009) denotes the g.f. as f(z) in Case B which is associated with F(t) the g.f. of A006077.

			G.f. = 1 + 3*q - 6*q^3 - 3*q^4 + 6*q^7 - 6*q^9 + 6*q^12 + 6*q^13 - 3*q^16 + ...

D. Zagier, Integral solutions of Apery-like recurrence equations, in: Groups and Symmetries: from Neolithic Scots to John McKay, CRM Proc. Lecture Notes 47, Amer. Math. Soc., Providence, RI, 2009, pp. 349-366.

Seiichi Manyama, 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
D. Zagier, Integral solutions of Apery-like recurrence equations.

Cf. A005928, A006077, A113062, A180318, A227454, A227696.

The Apéry-like numbers [or Apéry-like sequences, Apery-like numbers, Apery-like sequences] include A000172, A000984, A002893, A002895, A005258, A005259, A005260, A006077, A036917, A063007, A081085, A093388, A125143 (apart from signs), A143003, A143007, A143413, A143414, A143415, A143583, A183204, A214262, A219692, A226535, A227216, A227454, A229111 (apart from signs), A260667, A260832, A262177, A264541, A264542, A279619, A290575, A290576. (The term "Apery-like" is not well-defined.)

a[ n_] := SeriesCoefficient[ QPochhammer[ -q]^3 / QPochhammer[ -q^3], {q, 0, n}]

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^3 + A) * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A))^3, n))}

Expansion of f(q)^3 / f(q^3) in powers of q where f() is a Ramanujan theta function.
Expansion of 2*b(q^4) - b(q) = b(q^2)^3 / (b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of eta(q^2)^9 * eta(q^3) * eta(q^12) / (eta(q) * eta(q^4) * eta(q^6))^3 in powers of q.
Euler transform of period 12 sequence [ 3, -6, 2, -3, 3, -4, 3, -3, 2, -6, 3, -2, ...].
Moebius transform is period 36 sequence [ 3, -3, -9, -3, -3, 9, 3, 3, 0, 3, -3, 9, 3, -3, 9, -3, -3, 0, 3, 3, -9, 3, -3, -9, 3, -3, 0, -3, -3, -9, 3, 3, 9, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 972^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A227696.
G.f.: f(q) = F(t(q)) where F() is the g.f. of A006077 and t() is the g.f. of A227454.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^(3*k)).
a(3*n + 2) = a(4*n + 2) = 0.
a(n) = (-1)^n * A005928(n) = (-1)^(((n+1) mod 6 ) > 3) * A113062(n). A113062(n) = |a(n)|.
a(3*n) = A180318(n). a(2*n + 1) = 3 * A123530(n). a(4*n) = A005928(n).

A253243 Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Jun 04 2015

sign

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

			G.f. = 1 - 2*x^2 + 2*x^3 - 4*x^5 + x^6 + 2*x^9 + 3*x^12 - 4*x^14 + 2*x^15 + ...
G.f. = q - 2*q^9 + 2*q^13 - 4*q^21 + q^25 + 2*q^37 + 3*q^49 - 4*q^57 + ...

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. A005928, A097109, A112605, A112848, A123477, A123530, A123884, A226535, A246650.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x^2] QPochhammer[ -x^3, x^6] EllipticTheta[ 2, 0, x^(3/2)] / (2 x^(3/8)), {x, 0, n}];

{a(n) = if( n<0, 0, n = 4*n + 1; sumdiv(n, d, [ 0, 1, -1, -3, 1, -1, 3, 1, -1] [d%9 + 1]))};

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

Expansion q^(-1/4) * eta(q^2)^2 * eta(q^6)^4 / (eta(q^3)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, -2, 2, -1, 0, -4, 0, -1, 2, -2, 0, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246650.
a(n) = A123530(2*n) = A097109(4*n + 1) = A112848(4*n + 1) = A123477(4*n + 1). 3 * a(n) = A226535(4*n + 1). -3 * a(n) = A005928(4*n + 1).
a(3*n) = A123884(n). a(3*n + 1) = 0. a(3*n + 2) = -2 * A112605(n).

A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 04 2015

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A097195, A093829, A112848, A113447, A123530, A123863, A227696, A246838, A248897, A253243, A260941, A260942, A260943, A260944.

Cf. A000122, A000700, A004016, A010054, A005882, A005928, A121373.

A := Basis( ModularForms( Gamma1(36), 1), 80); A[2] + A[3] - 2*A[4] - A[5] - 2*A[7] + 2*A[8] + A[9] - 2*A[10] + 2*A[13] + 2*A[14] + 2*A[15] - A[17] - 2*A[19] - 4*A[20];

a[ n_] := If[ n < 1, 0, Sum[ {1, 1, 0, -1, -1, 0}[[Mod[ d, 6, 1]]] {1, 0, -2, 0, 1, 0}[[Mod[ n/d, 6, 1]]], {d, Divisors @ n}]]
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # == 2, -(-1)^#2, # == 3, -2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(9/2)] EllipticTheta[ 3, 0, q] / (2 q^(1/4) QPochhammer[ q^6]), {q, 0, n}];

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

{a(n) = 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, p%6==5, 1-e%2, e+1)))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^18 + A)), n))};

Expansion of (a(q) + a(q^2) - 3*a(q^3) - 2*a(q^4) - 3*a(q^6) + 6*a(q^12)) / 6 in powers of q where a() is a cubic AGM theta function.
Expansion of q * phi(q) * psi(-q) * psi(-q^9) / f(-q^6) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^4 * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^6) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, -3, 1, -2, 1, -2, 1, -2, 0, -3, 1, -1, 1, -3, 1, -2, 1, -2, 1, -2, 1, -3, 1, -1, 1, -3, 0, -2, 1, -2, 1, -2, 1, -3, 1, -2, ...].
Moebius transform is period 36 sequence [ 1, 0, -3, -2, -1, 0, 1, 2, 0, 0, -1, 6, 1, 0, 3, -2, -1, 0, 1, 2, -3, 0, -1, -6, 1, 0, 0, -2, -1, 0, 1, 2, 3, 0, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -(-1)^e if e>0, a(3^e) = -2, if e>0, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A123863.
a(2*n) = A112848(n). a(2*n + 1) = A123530(n). a(3*n) = -2 * A113447(n). a(3*n + 1) = A227696(n).
a(4*n) = - A112848(n). a(4*n + 1) = A253243(n). a(4*n + 2) = A123530(n). a(4*n + 3) = -2 * A246838(n).
a(6*n) = -2 * A093829(n). a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n). a(6*n + 3) = -2 * A033762(n). a(6*n + 5) = 0.
a(8*n + 1) = A260941(n). a(8*n + 2) = A253243(n). a(8*n + 3) = -2 * A260943(n). a(8*n + 4) = - A123530(n). a(8*n + 5) = 2 * A260942(n). a(8*n + 6) = -2 * A246838(n). a(8*n + 7) = 2 * A260944(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

cp's OEIS Frontend

A005928 G.f.: s(1)^3/s(3), where s(k) = eta(q^k) and eta(q) is Dedekind's function, cf. A010815.

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A226535 Expansion of b(-q) in powers of q where b() is a cubic AGM theta function.

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A253243 Expansion of phi(-x^2) * psi(x^3) * chi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

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A260945 Expansion of (2*b(q^4) - b(q) - b(q^2)) / 3 in powers of q where b() is a cubic AGM theta function.

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