A033762 -id:A033762 - cp's OEIS frontend

A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

1, 2, 4, 2, 2, 0, 4, 4, 4, 2, 0, 0, 2, 4, 8, 0, 2, 0, 4, 4, 0, 4, 0, 0, 4, 2, 8, 2, 4, 0, 0, 4, 4, 0, 0, 0, 2, 4, 8, 4, 0, 0, 8, 4, 0, 0, 0, 0, 2, 6, 4, 0, 4, 0, 4, 0, 8, 4, 0, 0, 0, 4, 8, 4, 2, 0, 0, 4, 0, 0, 0, 0, 4, 4, 8, 2, 4, 0, 8, 4, 0, 2, 0, 0, 4, 0, 8, 0, 0, 0, 0, 8, 0, 4, 0, 0, 4, 4, 12, 0, 2, 0, 0, 4, 8

Offset: 0

Michael Somos, Sep 26 2006

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

			G.f. = 1 + 2*q + 4*q^2 + 2*q^3 + 2*q^4 + 4*q^6 + 4*q^7 + 4*q^8 + 2*q^9 + ... - _Michael Somos_, Aug 11 2009

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, A107760, A112604, A112605, A113973, A121361, A123331, A123884, A275486.

Cf. A000700, A000122, A010054, A121373.

QP = QPochhammer; s = QP[q^2]*(QP[q^3]^6/(QP[q]^2*QP[q^6]^3)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

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

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

A = ModularForms( Gamma1(6), 1, prec=90).basis(); A[0] + 2*A[1] # Michael Somos, Sep 27 2013

Expansion of c(q)^2 / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
Expansion of phi(-x^3)^3 / phi(-x) where phi() is a Ramanujan theta function.
a(n) = 2*b(n) where b(n) is multiplicative and b(2^e) = (1 - 3*(-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
Euler transform of period 6 sequence [ 2, 1, -4, 1, 2, -2, ...].
Moebius transform is period 6 sequence [ 2, 2, 0, -2, -2, 0, ...].
a(n) = 2 * A123331(n) if n>0. (-1)^n * a(n) = A113973(n).
G.f.: Product_{k>0} (1 + x^k)/(1 - x^k) * ((1 - x^(3*k)) / (1 + x^(3*k)))^3.
G.f.: 1 + 2 * Sum_{k>0} x^k / (1 - x^k + x^(2*k)) = theta_3(-x^3)^3 / theta_3(-x).
From Michael Somos, Aug 11 2009: (Start)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v * (u - v)^2 - 2 * u * w * (v - w).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (16/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A107760.
a(4*n) = a(3*n) = a(n). a(12*n + 10) = a(6*n + 5) = 0.
a(2*n + 1) = 2 * A033762(n). a(3*n + 1) = 2 * A033687(n). a(4*n + 1) = 2 * A112604(n). a(4*n + 3) = 2 * A112605(n). a(6*n + 1) = 2 * A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 4 * A121361(n). (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 4*Pi/(3*sqrt(3)) = 2.418399... (A275486). - Amiram Eldar, Nov 14 2023

A132973 Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 07 2007

sign

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. = 1 - 3*q + 3*q^2 - 3*q^3 + 3*q^4 + 3*q^6 - 6*q^7 + 3*q^8 - 3*q^9 + 3*q^12 + ...

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. A033687, A097195, A107760, A132973, A132974, A227696.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]/2, {q, 0, n}]; (* Michael Somos, May 26 2013 *)

{a(n) = if( n<1, n==0, 3 * (-1)^n * sumdiv(n, d, kronecker(-12, d)))};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( 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 )), n))};

Expansion of b(q^2)^2 / b(-q) = b(q) * b(q^4) / b(q^2) in powers of q where b() is a cubic AGM theta function.
Expansion of (a(q^2) + 2 * a(q^4) - a(q)) / 2 = (c(q)^2 - 5 * c(q) * c(q^4) + 4 * c(q^4)^2) / (3 * c(q^2)) in powers of q where a(), c() are cubic AGM theta functions. - Michael Somos, May 26 2013
Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -3, 0, -2, -3, -3, 0, -3, -3, -2, 0, -3, -2, ...].
Moebius transform is period 12 sequence [ -3, 6, 0, 0, 3, 0, -3, 0, 0, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 108^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A113447.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^(2*k))^3 / ((1 - x^(3*k)) * (1 + x^(6*k))).
G.f.: 1 + 3 * Sum_{k>0} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: 1 + 3 * ( 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 * A107760(n). Convolution inverse of A132974.
a(2*n) = A107760(n). a(2*n + 1) = -3 * A033762(n). a(3*n) = A132973(n). a(3*n + 1) = -3 * A227696(n). - Michael Somos, Oct 31 2015
a(6*n + 1) = -3 * A097195(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0. - Michael Somos, Oct 31 2015

A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 8, 8, 1, 12, 12, 4, 14, 16, 6, 16, 18, 2, 20, 24, 8, 24, 24, 8, 31, 28, 1, 32, 30, 12, 32, 32, 12, 36, 48, 4, 38, 40, 14, 48, 42, 16, 44, 48, 6, 48, 48, 16, 57, 62, 18, 56, 54, 2, 72, 64, 20, 60, 60, 24, 62, 64, 8, 64, 84, 24, 68, 72, 24, 96, 72, 8, 74, 76, 31

Offset: 1

Michael Somos, Aug 21 2005, Apr 18 2007

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 + 2*q^2 + q^3 + 4*q^4 + 6*q^5 + 2*q^6 + 8*q^7 + 8*q^8 + q^9 + ...

Bruce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 223 Entry 3(iii).
Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 87, Eq. (33.2).

Seiichi Manyama, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033762, A131946, A222171.

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

a[ n_] := If[ n < 1, 0, Sum[ Mod[n/d, 2] d KroneckerSymbol[ 9, d], { d, Divisors[ n]}]]; (* Michael Somos, Sep 19 2013 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, Sep 19 2013 *)

{a(n) = if( n<1, 0, sumdiv(n, d, (n/d % 2) * d * (d%3>0)))};

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

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};

A = ModularForms( Gamma0(6), 2, prec=50) . basis();  A[1] + 2*A[2]; # Michael Somos, Sep 19 2013

Expansion of (1/3) * (b(q^2)^2 / b(q))* (c(q^2)^2 / c(q)) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q^2) * eta(q^6))^4 / (eta(q) * eta(q^3))^2 in powers of q.
Euler transform of period 6 sequence [ 2, -2, 4, -2, 2, -4, ...].
Multiplicative with a(2^e) = 2^e, a(3^e) = 1, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u*w * (u - 4*v) - v * (v - 4*w)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (3/4) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A131946. - Michael Somos, Sep 19 2013
G.f.: Sum_{k>0} k * x^k * (1 - x^(2*k))^2 / (1 - x^(6*k)) = x * Product_{k>0} ((1 + x^k) * (1 + x^(3*k)))^4 * ((1 - x^k) * (1 - x^(3*k)))^2.
a(3*n) = a(n), a(2*n) = 2 * a(n).
Convolution square of A033762. - Michael Somos, Sep 19 2013
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^s) * (1 - 1/3^(s-1)) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/24 = 0.411233... (A222171). (End)

A004046 Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

Original entry on oeis.org

1, 0, 0, 26208, 530712, 6368544, 47331648, 256864608, 1116087336, 4092877152, 12996075456, 37058557536, 96952754808, 232778774592, 526258264896, 1128148021728, 2286143305992, 4451523096384, 8386247967552, 15130902687264, 26614339616592, 45684687301344

Offset: 0

N. J. A. Sloane

nonn

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 729 (t/i)^12 f(t) where q = exp(2 Pi i t). - Michael Somos, Dec 21 2015

			G.f. = 1 + 26208*x^3 + 530712*x^4 + 6368544*x^5 + 47331648*x^6 + ...
G.f. = 1 + 26208*q^6 + 530712*q^8 + 6368544*q^10 + 47331648*q^12 + ...

N. J. A. Sloane, Seven Staggering Sequences, in Homage to a Pied Puzzler, E. Pegg Jr., A. H. Schoen and T. Rodgers (editors), A. K. Peters, Wellesley, MA, 2009, pp. 93-110.

Vaclav Kotesovec, Table of n, a(n) for n = 0..2000
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.
G. Nebe and N. J. A. Sloane, Home page for lattice
H.-G. Quebbemann, Modular lattices in Euclidean spaces, J. Number Theory, 54 (1995), 190-202.
N. J. A. Sloane, Seven Staggering Sequences.

Cf. A107657.

A := Basis( ModularForms( Gamma1(3), 12), 22);  A[1] + 26208*A[4] + 530712*A[5]; /* Michael Somos, Dec 21 2015 */

a[ n_] := With[ {U1 = QPochhammer[ q]^3, U3 = QPochhammer[ q^3]^3, U9 = QPochhammer[ q^9]^3}, With[ {z = ( 1 + 9 q U9/U1)^3}, SeriesCoefficient[ (U1^3/U3)^4 (27 z^4 - 72 z^3 + 64 z^2 + 16 z - 8) / 27, {q, 0, n}]]]; (* Michael Somos, Dec 25 2015 *)

th3 = sum(n=1,noo\2, 2*x^(4*n^2), 1+A);
th4 = sum(n=1,noo\2, (-1)^n*2*x^(4*n^2), 1+A);
th2 = sum(n=0,noo\2, 2*x^(4*n^2+4*n+1), A);
chk("th3^4 == th4^4+th2^4");
/* A004016(x^4) */
phi0 = th2*subst(th2,x,x^3)+ th3*subst(th3,x,x^3);
/* 2*x*A033762(x^2) */
phi1 = th2*subst(th3,x,x^3)+ th3*subst(th2,x,x^3);
/* A004010(x^2) */
K_12 = phi0^6+45*phi0^2*phi1^4+18*phi1^6;
a=phi0;b=phi1;
A004046=a^12-9/2*a^8*b^4+414*a^6*b^6+1458*a^4*b^8+1998*a^2*b^10+459/2*b^12;

{a(n) = my(A, U1, U3, U9, z); if( n<0, 0, A = x * O(x^n); U1 = eta(x + A)^3; U3 = eta(x^3 + A)^3; U9 = eta(x^9 + A)^3; z = (1 + 9 * x * U9/U1)^3; polcoeff( (U1^3/U3)^4 * (27*z^4 - 72*z^3 + 64*z^2 + 16*z - 8) / 27, n))}; /* Michael Somos, Dec 25 2015 */

Theta series = a^12 - 9/2*a^8*b^4 + 414*a^6*b^6 + 1458*a^4*b^8 + 1998*a^2*b^10 + 459/2*b^12 (see PARI code for details).

G.f.: (27*a(x)^12 - 72*a(x)^9*b(x)^3 + 64*a(x)^6*b(x)^6 + 16*a(x)^3*b(x)^9 - 8*b(x)^12) / 27 where a(), b() are cubic AGM theta functions, - Michael Somos, Dec 25 2015

PARI code from Michael Somos, Jun 07 2005

A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, -2, 6, 0, 0, -4, 0, -2, 0, 0, 6, -4, 0, 0, 6, 0, 0, -4, 0, -4, 0, 0, 0, -2, 0, -2, 12, 0, 0, -4, 0, 0, 0, 0, 6, -4, 0, -4, 0, 0, 0, -4, 0, 0, 0, 0, 6, -6, 0, 0, 12, 0, 0, 0, 0, -4, 0, 0, 0, -4, 0, -4, 6, 0, 0, -4, 0, 0, 0, 0, 0, -4, 0, -2, 12, 0, 0, -4, 0, -2, 0, 0, 12, 0, 0, 0, 0, 0, 0, -8, 0, -4, 0, 0, 0, -4, 0, 0, 6, 0

Offset: 0

Michael Somos, Feb 09 2006

sign

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. = 1 - 2*q - 2*q^3 + 6*q^4 - 4*q^7 - 2*q^9 + 6*q^12 - 4*q^13 + ...

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. A004016, A033716, A033762, A097195, A112604, A122861.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Nov 09 2013 *)

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

{a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], if(p = A[k,1], e = A[k,2]; if( p==2, -3 * ((e+1)%2), if( p==3, 1, if( p%6==1, e+1, (e+1)%2))))))} /* Michael Somos, Nov 09 2013 */

Expansion of theta_4(q) * theta_4(q^3) in powers of q.
Expansion of (4 * a(q^4) - a(q)) / 3 = (4 * b(q^4) - b(q)) * b(q) / (3 * b(q^2)) in powers of q where a(), b() are cubic AGM theta functions. - Michael Somos, Nov 09 2013
Expansion of (eta(q) * eta(q^3))^2 / (eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -2, -1, -4, -1, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 2, 0, 6, 2, 0, -2, -6, 0, -2, 2, 0, ...]. - Michael Somos, Nov 09 2013
a(n) = -2*b(n) where b(n) is multiplicative and b(2^e) = -3 * (1 + (-1)^e) / 2 if e>0, b(3^e) = 1, b(p^e) = 1+e if p == 1 (mod 6), b(p^e) = (1 +(-1)^e) / 2 if p == 5 (mod 6).
Given g.f. A(x), then B(x) = A(x)^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v*(u + v)^2 - 4*u * (w^2 - v*w + v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 192^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A033762. - Michael Somos, Nov 09 2013
G.f.: 1 - 2*(Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - 4 * x^(4*k) / (1 + x^(4*k) + x^(8*k))).
G.f.: (Sum_{k in Z} (-x)^(k^2)) * (Sum_{k in Z} (-x)^(3*k^2)).
a(n) = -2 * A115979(n) unless n=0. a(n) = (-1)^n * A033716(n).
a(3*n + 2) = a(4*n + 2) = 0. a(3*n) = a(n). a(2*n + 1) = -2 * A033762(n). a(3*n + 1) = -2 * A122861(n). a(4*n) = A004016(n). a(4*n + 1) = -2 * A112604(n). a(6*n + 1) = -2 * A097195(n). - Michael Somos, Nov 09 2013

A123530 Expansion of q^(-1/2)eta(q)^2eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 02 2006

sign

G. C. Greubel, Table of n, a(n) for n = 0..1000

QP = QPochhammer; s = QP[q]^2*(QP[q^6]^3/(QP[q^2]*QP[q^3]^2)) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)

{a(n)=if(n<0, 0, n=2*n+1; sumdiv(n, d, kronecker(-12,d)*[0,1,0,-2,0,1][n/d%6+1]))}

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

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

Euler transform of period 6 sequence [ -2, -1, 0, -1, -2, -2, ...].
a(n) = b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(3^e) = -2 if e>0, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} F(x^(6k-5))-F(x^(6k-1)) where F(x)=(x-x^3)/(1+x^2+x^4).
a(3*n+2) = 0.
a(3*n) = A097195(n).
a(3*n+1) = -2*A033762(n).
a(n) = A097109(2*n+1) = A112848(2*n+1).

A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 1, 3, 0, -3, 2, -3, 1, 0, 0, 3, 2, -6, 0, 3, 0, -3, 2, 0, 2, 0, 0, -3, 1, -6, 1, 6, 0, 0, 2, -3, 0, 0, 0, 3, 2, -6, 2, 0, 0, -6, 2, 0, 0, 0, 0, 3, 3, -3, 0, 6, 0, -3, 0, -6, 2, 0, 0, 0, 2, -6, 2, 3, 0, 0, 2, 0, 0, 0, 0, -3, 2, -6, 1, 6, 0, -6, 2, 0, 1, 0, 0, 6, 0, -6, 0, 0, 0, 0, 4, 0, 2, 0, 0, -3, 2, -9, 0, 3, 0, 0, 2, -6, 0

Offset: 1

Michael Somos, Sep 02 2005

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

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

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

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. A033687, A033762, A093602, A093829, A097195, A112604, A112605, A129576, A244375.

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

A := Basis( ModularForms( Gamma1(12), 1), 106); A[2] - 3*A[3] + A[4] + 3*A[5]; /* Michael Somos, Jan 17 2015 */

a[ n_] := SeriesCoefficient[ q QPochhammer[ q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, 0, q^(3/2)] / (2 q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, JacobiSymbol[ -3, n/#] {1, -2, 1, 0}[[Mod[#, 4, 1]]] &]]; (* Michael Somos, Jan 17 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-3, n/d)*[0, 1, -2, 1][d%4 + 1]))};

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

From Michael Somos, Jan 17 2015: (Start)
Expansion of b(q) * (b(q^4) - b(q)) / (3*b(q^2)) in powers of q  where b() is a cubic AGM theta function.
Expansion of q * chi(-q)^3 * phi(-q^2) * psi(q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q)^2 * psi(q^6)^2 / (psi(-q) * psi(-q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * f(q) * f(-q, -q^5)^4 / f(q^3)^3 in powers of q where f() is a Ramanujan theta function. (End)
Expansion of (eta(q) * eta(q^12))^3 / (eta(q^2) * eta(q^3) * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ -3, -2, -2, -1, -3, 0, -3, -1, -2, -2, -3, -2, ...].
Moebius transform is period 12 sequence [ 1, -4, 0, 6, -1, 0, 1, -6, 0, 4, -1, 0, ...].
Multiplicative with a(2^e) = 3(-1)^e if 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 == 2 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(-3, k) * x^k * (1 - x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A244375(n). a(6*n + 5) = 0, a(3*n) = a(n).
a(2*n) = -3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n + 1) = A129576(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 2) = -3 * A033687(n).
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 22 2005

G. C. Greubel, Table of n, a(n) for n = 1..1000

Cf. A033687, A033762, A092829, A093829, A097195, A248897, A255648 (absolute values).

QP = QPochhammer; s = QP[q]*QP[q^2]*(QP[q^18]^2/(QP[q^6]*QP[q^9])) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (-1)^e; f[3, e_]:= -2; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 28 2024 *)

{a(n)=if(n<1, 0, if(n%3==0, n/=3; -2,1)* sumdiv(n,d,kronecker(-12,d) -if(d%2==0, 2*kronecker(-3,d/2))))}

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

Euler transform of period 18 sequence [ -1, -2, -1, -2, -1, -1, -1, -2, 0, -2, -1, -1, -1, -2, -1, -2, -1, -2, ...].
Moebius transform is period 18 sequence [1, -2, -3, 2, -1, 6, 1, -2, 0, 2, -1, -6, 1, -2, 3, 2, -1, 0, ...].
Multiplicative with a(2^e) = (-1)^e, 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.: Sum_{k>0} Kronecker(-3, k) x^k(1-x^(2k))^2/(1-x^(6k)) = x Product_{k>0} (1-x^k)(1-x^(2k))(1+x^(9k))(1+x^(6k)+x^(12k)).
a(3n) = -2*A092829(n). a(3n+1) = A093829(3n+1) = A033687(n). a(3n+2) = A093829(3n+2). a(6n)/2 = A093829(n). a(6n+1) = A097195(n). a(6n+3) = -2*A033762(n). a(6n+5) = 0.
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Jan 23 2024

A244375 Expansion of (a(q) + 3a(q^2) - 4a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jun 26 2014

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

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. A033687, A033762, A093602, A093829, A112298, A122861, A244339.

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

A := Basis( ModularForms( Gamma1(12), 1), 82); A[2] + 3*A[3] + A[4] - 3*A[5]; /* Michael Somos, Jan 17 2015 */

a[ n_] := If[ n < 1, 0, Sum[ Mod[ n/d, 2] {1, 3, 0, -3, -1, 0}[[ Mod[ d, 6, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^8 QPochhammer[ q^3] QPochhammer[ q^12]^4 / (QPochhammer[ q]^3 QPochhammer[ q^4]^4 QPochhammer[ q^6]^4), {q, 0, n}];
a[ n_] := SeriesCoefficient[ q QPochhammer[ -q, q^2]^3 QPochhammer[ -q^6, q^6]^3 EllipticTheta[ 4, 0, q^2] EllipticTheta[ 2, Pi/4, q^(3/2)] / (2^(1/2) q^(3/8)), {q, 0, n}]; (* Michael Somos, Jan 17 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, (n/d%2) * [0, 1, 3, 0, -3, -1][d%6 + 1]))};

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^3 + A) * eta(x^12 + A)^4 / (eta(x + A)^3 * eta(x^4 + A)^4 * eta(x^6 + A)^4), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)) * [0, 1, 4, 1][k%4 + 1], x * O(x^n)), n))};

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 - x^(2*k)) * [0, 1, 3, 0, -3, -1][k%6 + 1], x * O(x^n)), n))};

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

Expansion of (b(q) - b(q^4)) * (b(q) - 2*b(q^4)) / (3* b(q^2)) = b(q^2)^2 * (b(q^4) - b(q)) / (3 * b(q) * b(q^4)) in powers of q where b() is a cubic AGM theta function.
Expansion of q * phi(q)^2 * psi(q^6)^2 / (psi(q) * psi(q^3)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of q * chi(q)^3 * phi(-q^2) * psi(-q^3) / chi(-q^6)^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions. - Michael Somos, Jan 17 2015
Expansion of q * f(-q) * f(q, q^5)^4 / f(-q^3)^3 in powers of q where f() is a Ramanujan theta function. - Michael Somos, Jan 17 2015
Expansion of eta(q^2)^8 * eta(q^3) * eta(q^12)^4 / (eta(q)^3 * eta(q^4)^4 * eta(q^6)^4) in powers of q. - Michael Somos, Jan 17 2015
a(n) is multiplicative with a(2^e) = 3 * (-1)^(e+1) if 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).
Euler transform of period 12 sequence [ 3, -5, 2, -1, 3, -2, 3, -1, 2, -5, 3, -2, ...].
Moebius transform is period 12 sequence [ 1, 2, 0, -6, -1, 0, 1, 6, 0, -2, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A244339.
a(2*n) = 3 * A093829(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A122861(n). a(6*n + 2) = 3 * A033687(n). a(6*n + 5) = 0.
a(n) = -(-1)^n * A112298(n). - Michael Somos, Jan 17 2015
Sum_{k=1..n} abs(a(k)) ~ (Pi/sqrt(3)) * n. - Amiram Eldar, Jan 23 2024

A005881 Theta series of planar hexagonal lattice (A2) with respect to edge.

Original entry on oeis.org

2, 2, 0, 4, 2, 0, 4, 0, 0, 4, 4, 0, 2, 2, 0, 4, 0, 0, 4, 4, 0, 4, 0, 0, 6, 0, 0, 0, 4, 0, 4, 4, 0, 4, 0, 0, 4, 2, 0, 4, 2, 0, 0, 0, 0, 8, 4, 0, 4, 0, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 2, 0, 0, 4, 4, 0, 8, 0, 0, 4, 0, 0, 0, 6, 0, 4, 0, 0, 4, 0, 0, 4, 0, 0, 6, 4, 0, 4, 0, 0, 4, 4, 0, 0, 4, 0, 4, 0, 0, 4, 4, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

Also number of ways of writing n as the sum of a triangular number and three times a triangular number.
The hexagonal lattice is the familiar 2-dimensional lattice in which each point has 6 neighbors. This is sometimes called the triangular lattice.
Given g.f. A(x), then q^(1/2)*A(q) is denoted phi_1(z) where q=exp(Pi*i*z) in Conway and Sloane.
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Antti Karttunen, Table of n, a(n) for n = 0..10000
J. H. Conway and N. J. A. Sloane, Sphere Packings, Lattices and Groups, Springer-Verlag, p. 103. see Equ. (13).
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
G. Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2
N. J. A. Sloane, Theta series and magic numbers for diamond and certain ionic crystal structures, J. Math. Phys. 28 (1987), 1653-1657.
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.

Cf. A033762.

d:=proc(r,m,n) local i,t1; t1:=0; for i from 1 to n do if n mod i = 0 and i-r mod m = 0 then t1:=t1+1; fi; od: t1; end; [seq(2*(d(1,3,2*n+1)-d(2,3,2*n+1)),n=0..120)];

a[n_] := 2*DivisorSum[2n+1, KroneckerSymbol[-12, #]*Mod[(2n+1)/#, 2]& ]; Table[a[n], {n, 0, 105}] (* Jean-François Alcover, Dec 02 2015, adapted from PARI *)

{a(n) = if( n<0, 0, n = 2*n + 1; 2 * sumdiv(n, d, kronecker( -12, d) * (n/d%2)))}; /* Michael Somos, Nov 05 2006 */

{a(n) = if( n<0, 0, n = 8*n + 4; 2 * sum(j=1, sqrtint(n\3), (j%2) * issquare(n - 3*j^2)))}; /* Michael Somos, Nov 05 2006 */

Expansion of q^(-1) * (a(q) - a(q^4)) / 3 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Nov 05 2006

a(n) = 2*A033762(n).

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A123330 Expansion of eta(q^2) * eta(q^3)^6 / (eta(q)^2 * eta(q^6)^3) in powers of q.

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A132973 Expansion of psi(-q)^3 / psi(-q^3) in powers of q where psi() is a Ramanujan theta function.

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A111932 Expansion of q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function.

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A004046 Theta series of extremal 3-modular even 24-dimensional lattice with minimal norm 6 and det = 3^12.

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A115978 Expansion of phi(-q) * phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A123530 Expansion of q^(-1/2)*eta(q)^2*eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

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A123530 Expansion of q^(-1/2)eta(q)^2eta(q^6)^3/(eta(q^2)*eta(q^3)^2) in powers of q.

A112298 Expansion of (a(q) - 3a(q^2) + 2a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.

A112848 Expansion of eta(q)eta(q^2)eta(q^18)^2/(eta(q^6)*eta(q^9)) in powers of q.

A244375 Expansion of (a(q) + 3a(q^2) - 4a(q^4)) / 6 in powers of q where a() is a cubic AGM theta function.