A121361 -id:A121361 - cp's OEIS frontend

A112605 Number of representations of n as a sum of a square and six times a triangular number.

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

Offset: 0

James Sellers, Dec 21 2005

nonn

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			a(22) = 4 since we can write 22 = 4^2 + 6*1 = (-4)^2 + 6*1 = 2^2 + 6*3 = (-2)^2 + 6*3.
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 + ... - _Michael Somos_, Aug 11 2009
G.f. = q^3 + 2*q^7 + 2*q^19 + q^27 + 2*q^31 + 2*q^39 + 2*q^43 + 2*q^63 + ... - _Michael Somos_, Aug 11 2009

G. C. Greubel, Table of n, a(n) for n = 0..1000
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A112608(n) = a(2*n). 2 * A112609(n) = a(2*n + 1). A112604(n) = a(3*n). 2 * A121361(n) = a(3*n + 1). A112606(n) = a(6*n). 2 * A131962(n) = a(6*n + 1). 2 * A112607(n) = a(6*n + 3). 2 * A131964(n) = a(6*n + 4). - Michael Somos, Aug 11 2009

a[n_] := DivisorSum[4n+3, KroneckerSymbol[-3, #]&]; Table[a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

{a(n) = if(n<0, 0, sumdiv(4*n+3, d, kronecker(-3, d)))}; /* Michael Somos, May 20 2006 */

{a(n) = my(A); if(n<0, 0, A = x*O(x^n); polcoeff( eta(x^2+A)^5*eta(x^12+A)^2 / eta(x+A)^2 / eta(x^4+A)^2 / eta(x^6+A), n))}; /* Michael Somos, May 20 2006 */

a(n) = d_{1, 3}(4n+3) - d_{2, 3}(4n+3) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-3/4)*eta(q^2)^5*eta(q^12)^2/(eta(q)^2*eta(q^4)^2*eta(q^6)) in powers of q. - Michael Somos, May 20 2006
Euler transform of period 12 sequence [ 2, -3, 2, -1, 2, -2, 2, -1, 2, -3, 2, -2, ...]. - Michael Somos, May 20 2006
a(n)=A002324(4n+3). - Michael Somos, May 20 2006
Expansion of phi(q)*psi(q^6) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, May 20 2006, Sep 29 2006
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 3^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A164273. - Michael Somos, Aug 11 2009
a(3*n + 2) = 0. - Michael Somos, Aug 11 2009

A035178 a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).

Original entry on oeis.org

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

N. J. A. Sloane

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

			G.f. = q + q^2 + q^3 + q^4 + q^6 + 2*q^7 + q^8 + q^9 + q^12 + 2*q^13 + 2*q^14 + ...

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 346.

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, A093829, A093766, A097197, A107760, A112604, A112605.
Cf. A112606, A112607, A112608, A121361, A123884, A131961, A131962.
Cf. A131963, A131964, A137608.
Cf. A000122, A000700, A010054, A121373.

A := Basis( ModularForms( Gamma1(6), 1), 88); B := (A[1] - 1) / 3 + A[2]; B; /* Michael Somos, Aug 04 2015 */

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -12, d], { d, Divisors[ n]}]]; (* Michael Somos, Jun 24 2011 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, 1, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1 ] & @@@ FactorInteger@n)]; (* Michael Somos, Aug 04 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(1/2)]^3 / EllipticTheta[ 2, 0, q^(3/2)] - 4) / 12, {q, 0, n}]; (* Michael Somos, Aug 04 2015 *)
a[n_] := DivisorSum[n, KroneckerSymbol[-12, #]&]; Array[a, 105] (* Jean-François Alcover, Dec 01 2015 *)

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -12, d)))}; /* Michael Somos, Apr 18 2004 */

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -12, p) * X))) [n])}; /* Michael Somos, Jun 24 2011 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^3 + A) * eta(x^2 + A)^6 / (eta(x^6 + A)^2 * eta(x + A)^3) - 1) / 3, n))}; /* Michael Somos, Aug 11 2009 */

{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<5, 1, p%6==5, 1-e%2, 1+e)))}; /* Michael Somos, Aug 04 2015 */

Moebius transform is period 6 sequence [ 1, 0, 0, 0, -1, 0, ...]. - Michael Somos, Feb 14 2006
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) * (u1 - u2 - u3 + u6) - (u2 -u6) * (1 + 3*u6). - Michael Somos, May 29 2005
Dirichlet g.f.: zeta(s) * L(chi,s) where chi(n) = Kronecker( -12, n). Sum_{n>0} a(n) / n^s = Product_{p prime} 1 / ((1 - p^-s) * (1 - Kronecker( -12, p) * p^-s)). - Michael Somos, Jun 24 2011
a(n) is multiplicative with a(p^e) = 1 if p=2 or p=3, a(p^e) = 1+e if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
G.f.: Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) = Sum_{k>=0} x^(6*k + 1) / (1 - x^(6*k + 1)) - x^(6*k + 5) / (1 - x^(6*k + 5)). - Michael Somos, Feb 14 2006
a(n) = |A093829(n)| = -(-1)^n * A137608(n) = a(2*n) = a(3*n).  a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
From Michael Somos, Aug 11 2009: (Start)
3 * a(n) = A107760(n) unless n=0. a(2*n + 1) = A033762(n). a(3*n + 1) = A033687(n). a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 7) = 2 * A112608(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). (End)
Expansion of (psi(q)^3 / psi(q^3) - 1) / 3 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Aug 04 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 16 2023

Definition edited by Michael Somos, Aug 11 2009

A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Sep 28 2006, Apr 04 2008

Expansion of (a(q) - 2 * a(q^2) - a(q^4) + 2*a(q^8)) / 6 in powers of q where a() is a cubic AGM function.

			q - 2*q^2 + q^3 - 2*q^6 + 2*q^7 + q^9 + 2*q^13 - 4*q^14 - 2*q^18 + ...

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

A033762(n) = a(2*n+1). A112604(n) = a(4*n+1). -2 * A033762(n) = a(4*n+2). A112605(n) = a(4*n+3). A097195(n) = a(6*n+1). A112606(n) = a(8*n+1). -2 * A112604(n) = a(8*n+2). A112608(n) = a(8*n+3). 2 * A112607(n) = a(8*n+5). -2 * A112605(n) = a(8*n+6). 2 * A112609(n) = a(8*n+7).
A123884(n) = a(12*n+1). 2 * A121361(n) = a(12*n+7). A131961(n) = a(24*n+1). 2 * A131962(n) = a(24*n+7). A112608(n) = a(24*n+9). 2 * A131963(n) = a(24*n+13). 2 * A131964(n) = a(24*n+19).
Cf. A093766, A136748.

QP = QPochhammer; s = QP[x]^2*QP[x^6]^4*QP[x^8]*(QP[x^24]/(QP[x^2]*QP[x^3]* QP[x^12])^2) + O[x]^105; CoefficientList[s, x] (* Jean-François Alcover, Nov 06 2015, adapted from PARI, updated Dec 06 2015 *)

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

{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, -2*(e<2), if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^8 + A) * eta(x^24 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))}

Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -2, -2, -1, 0, 0, -2, 0, -2, 0, 0, -1, -2, -2, -2, 0, 0, 0, -2, -2, ...].
Moebius transform is period 24 sequence [ 1, -3, 0, 2, -1, 0, 1, 0, 0, 3, -1, 0, 1, -3, 0, 0, -1, 0, 1, -2, 0, 3, -1, 0, ...].
a(n) is multiplicative with a(2) = -2, a(2^e) = 0 if e>1, 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. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A136748.
G.f.: x * Product_{k>0} (1 -x^(6*k)) * (1 - x^k + x^(2*k))^2 * (1 - x^(8*k)) * (1 + x^(12*k)) / (1 + x^(6*k)).
a(4*n) = a(6*n + 4) = a(6*n + 5) = 0. a(3*n) = a(n).
Sum_{k=1..n} abs(a(k)) ~ c * n, where c = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Jan 22 2024

A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -6, 12, -6, -6, 0, 12, -12, 12, -6, 0, 0, -6, -12, 24, 0, -6, 0, 12, -12, 0, -12, 0, 0, 12, -6, 24, -6, -12, 0, 0, -12, 12, 0, 0, 0, -6, -12, 24, -12, 0, 0, 24, -12, 0, 0, 0, 0, -6, -18, 12, 0, -12, 0, 12, 0, 24, -12, 0, 0, 0, -12, 24, -12, -6, 0, 0, -12, 0, 0, 0, 0, 12, -12, 24, -6, -12, 0, 24, -12, 0, -6, 0, 0, -12, 0, 24

Offset: 0

Michael Somos, Sep 15 2006

sign

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

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

Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.64).

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, A112604, A112605, A112606, A112607, A112608, A112609, A113660, A121361, A122860, A123884, A227354.

Cf. A000122, A000700, A010054, A121373.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q]^3 / EllipticTheta[ 4, 0, q^3], {q, 0, n}] (* Michael Somos, Sep 27 2013 *)

{a(n)= if( n<1, n==0, 6 * sumdiv(n, d, (-1)^(n/d) * kronecker( -3, d)))}

{a(n)= if( n<1, n==0, -6 * sumdiv(n, d, (2 + (-1)^d) * kronecker( -3, d)))}

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

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

Expansion of 2*a(q^2) - a(q) = b(q)^2 / b(q^2) in powers of q where a(), b() are cubic AGM theta functions.
Expansion of eta(q)^6 * eta(q^6) / (eta(q^2)^3 * eta(q^3)^2) in powers of q.
Euler transform of period 6 sequence [ -6, -3, -4, -3, -6, -2, ...].
Moebius transform is period 6 sequence [ -6, 18, 0, -18, 6, 0, ...].
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. 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+u6) * (u1+2*u2+u3) - (2*u1+u2-2*u3-u6) * (u1+2*u2-u3).
G.f.: Product_{k>0} (1 + x^(3*k)) / (1 + x^k)^3 * (1 - x^k)^3 / (1 - x^(3k)) = 1 + 6 * Sum_{k>0} (-1)^k * x^k / (1 + x^k + x^(2*k)).
G.f.: 1 - 6 * (Sum_{k>0} x^(3*k - 2) / (1 + x^(3*k - 2)) - x(3*k - 1)
/ (1 + x^(3*k - 1))).
a(3*n) = a(4*n) = a(n). a(6*n + 5) = 0.
(-1)^n * a(n) = A113660(n). -6 * a(n) = A122860(n) if n>0.
a(2*n) = A227354(n). a(2*n + 1) = -6 * A033762(n). a(3*n + 1) = -6 * A033687(n). a(4*n + 1) = -6 * A112604(n). a(4*n + 3) = -6 * A112605(n). a(6*n + 1) = -6 * A097195(n). a(8*n + 1) = -6 * A112606(n). a(8*n + 3) = -6 * A112608(n). a(8*n + 5) = -12 * A112607(n-1). a(8*n + 7) = -12 * A112609(n). a(12*n + 1) = -6 * A123884(n). a(12*n + 7) = -12 * A121361(n). - Michael Somos, Sep 27 2013
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 0. - Amiram Eldar, Nov 23 2023

A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

Original entry on oeis.org

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, Nov 02 2005

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
Li-Chien Shen, On the Modular Equations of Degree 3, Proc. Amer. Math. Soc. 122 (1994), no. 4, 1101-1114. See p. 1105, equation (3.8).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A033687, A033762, A035178, A073010, A093829, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964. A133637.

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

A := Basis( ModularForms( Gamma1(24), 1), 106); A[2] + A[3] + A[4] - A[5] + A[7] + 2*A[8] + A[9] + A[10]; /* Michael Somos, May 07 2015 */

a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, Jan 31 2015 *)

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

{a(n) = if( n<1, 0, direuler( p=2, n, if( p==2, 1 + X / (1 + X), 1 / ((1 - X) * (1 - kronecker( -12, p) * X))))[n])};

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

{a(n) = if( n<1, 0, sumdiv(n, d, [ 0, 1, 0, 0, -2, -1, 0, 1, 2, 0, 0,-1][d%12 + 1]))}; /* Michael Somos, May 07 2015 */

Expansion of (eta(q^2) * eta(q^3)^3 * eta(q^12)^3) / (eta(q) * eta(q^4) * eta(q^6)^3) in powers of q.
Euler transform of period 12 sequence [1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -2, ...].
Moebius transform is period 12 sequence [1, 0, 0, -2, -1, 0, 1, 2, 0, 0, -1, 0, ...].
a(n) is multiplicative and a(2^e) = -(-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 == 5 (mod 6).
G.f.: Sum_{k>0} x^(6*k - 5) / (1 - x^(6*k - 5)) - x^(6*k - 1) / (1 - x^(6*k - 1)) - 2 * x^(12*k - 8) / (1 - x^(12*k - 8)) + 2 * x^(12*k - 4) / (1 - x^(12*k-4)).
G.f.: Sum_{k>0} x^k * (1 - x^(3*k))^2 / (1 + x^(4*k) + x^(8*k)).
G.f.: x * Product_{k>0} (1 - x^k) / (1 - x^(4*k - 2)) * ((1 - x^(12*k - 6)) / (1 - x^(3*k)))^3.
Expansion of theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.
Expansion of q * psi(-q^3)^3 / psi(-q) in powers of q where psi() is a Ramanujan theta function.
Expansion of (c(q) * c(q^4)) / (3 * c(q^2)) in powers of q where c() is a cubic AGM theta function.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = (4/3)^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A132973.
a(n) = -(-1)^n * A093829(n). - Michael Somos, Jan 31 2015
Convolution inverse of A133637.
a(3*n) = a(n). a(6*n + 5) = a(12*n + 10) = 0. |a(n)| = A035178(n).
a(2*n) = A093829(n). a(2*n + 1) = A033762(n).
a(4*n + 1) = A112604(n). a(4*n + 3) = A112605(n).
a(6*n + 1) = A097195(n). a(6*n + 2) = A033687(n).
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n). a(8*n + 6) = A112605(n). 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 * A121963(n). a(24*n + 19) = 2 * A131964(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(6*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 23 2023

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

Original entry on oeis.org

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

A227696 Expansion of f(x^3)^3 / f(x) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 22 2013

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^4 - x^5 + 2*x^6 + x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 + 2*q^13 - q^16 + 2*q^19 + q^25 - 2*q^28 + 2*q^31 + ...

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, A121361, A123884, A226535.

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

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

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^6)^9 / (eta(q^2) * eta(q^3) * eta(q^12))^3 in powers of q.
Euler transform of period 12 sequence [ -1, 2, 2, 1, -1, -4, -1, 1, 2, 2, -1, -2, ...].
Moebius transform is period 36 sequence [ 1, -1, -1, -1, -1, 1, 1, 1, 0, 1, -1, 1, 1, -1, 1, -1, -1, 0, 1, 1, -1, 1, -1, -1, 1, -1, 0, -1, -1, -1, 1, 1, 1, 1, -1, 0, ...].
a(n) = b(3*n+1) where b(n) is multiplicative with b(2^e) = - (1 + (-1)^e) / 2 if e>0, b(3^e) = 0^e, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1 + (-1)^3) / 2 if p == 5 (mod 6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (4/3)^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A226535.
G.f.: Product_{k>0} (1 - (-x)^k)^3 / (1 - (-x)^k).
a(n) = (-1)^n * A033687(n). a(4*n + 3) = 0.
a(2*n) = A097195(n). a(4*n) = A123884(n). a(4*n + 1) = - A033687(n). a(4*n + 2) = 2 * A121361(n).

A138806 Expansion of (theta_3(q) * theta_3(q^27) + theta_2(q) * theta_2(q^27) - 1) / 2 in powers of q.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Mar 30 2008

Half the number of integer solutions to x^2 + x*y + 7*y^2 = n. - Jianing Song, Nov 20 2019

			q + q^4 + 2*q^7 + 3*q^9 + 2*q^13 + q^16 + 2*q^19 + q^25 + 3*q^27 + ...

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A138805 (number of integer solutions to x^2 + x*y + 7*y^2 = n).
Cf. A033687, A073010, A097195, A123884, A121361.
Similar sequences: A096936, A113406, A110399.

f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := 1 - Mod[e, 2]; f[3, e_] := 3; f[3, 1] = 0; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 07 2023 *)

{a(n) = if( n<1, 0, if( n%3 == 2, 0, if( n%3==1, sumdiv(n, d, kronecker(-3, d)), if( n%9==0, 3 * sumdiv(n/9, d, kronecker(-3, d))))))}

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

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 14], n, 1)[n])}

a(n) is multiplicative and a(3^e) = 3 if e>1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 6).
a(3*n + 2) = a(4*n + 2) = 0.
G.f.: (Sum_{i,j} x^(i*i + i*j + 7*j*j) - 1) / 2.
A138805(n) = 2 * a(n) unless n=0. A033687(n) = a(3*n + 1). A097195(n) = a(6*n + 1). A123884(n) = a(12*n + 1). 2 * A121361(n) = a(12*n + 7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 16 2023

A246838 Expansion of f(-x^2) * f(-x^12)^2 / f(x^1, x^5) in powers of x where f() is Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Sep 04 2014

sign

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

			G.f. = 1 - x - x^4 + x^6 - x^7 + 2*x^9 - x^10 + 2*x^15 - x^16 + x^18
+ ...
G.f. = q^3 - q^7 - q^19 + q^27 - q^31 + 2*q^39 - q^43 + 2*q^63 - q^67 + ...

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. A112604, A121361, A246752.

a[ n_] := SeriesCoefficient[ x^(1/4) EllipticTheta[ 2, Pi/4, x^(1/2)] QPochhammer[ x^12]^2 / EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Aug 27 2015 *)

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

Expansion of q^(-3/4) * eta(q) * eta(q^4) * eta(q^6) * eta(q^12) / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 27^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246752.
a(3*n) = A112604(n). a(3*n + 1) = - A121361(n). a(3*n + 2) = 0.

A123863 Expansion of (c(q^3) - c(q^6) - 2*c(q^12)) / 3 in powers of q where c(q) is a cubic AGM theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Oct 14 2006

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^4 + 2*q^7 - q^8 + 2*q^13 - 2*q^14 - q^16 + 2*q^19 + ...

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. A033687, A097195, A113448, A121361, A123884.

a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 4, {1, -1, 0}[[Mod[#, 3, 1]]], Mod[#, 6] == 1, #2 + 1, True, (1 + (-1)^#2) / 2] & @@@ FactorInteger @ n)]; (* Michael Somos, Aug 03 2015 *)
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 2, 0, x^(9/2)] EllipticTheta[ 2, Pi/4, x^(1/2)] EllipticTheta[ 4, 0, x^18] / (2^(3/2) x^(5/4) QPochhammer[ x^6]), {x, 0, n}]; (* Michael Somos, Aug 03 2015 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^4 / (eta(x^2 + A) * eta(x^6 + A) * eta(x^9 + A) * eta(x^36 + A)), 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, p==3, 0, p%6==1, e+1, !(e%2))))};

Expansion of (a(x) - a(x^2) - a(x^3) - 2*a(x^4) + a(x^6) + 2*a(x^12)) / 6 in powers of x where a() is a cubic AGM theta function. - Michael Somos, Aug 03 2015
Expansion of psi(-x) * psi(-x^9) * phi(x^9) / f(-x^6) in powers of x where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Aug 03 2015
Expansion of eta(q) * eta(q^4) * eta(q^18)^4 / (eta(q^2) * eta(q^6) * eta(q^9) * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -1, 0, -1, -1, -1, 1, -1, -1, 0, 0, -1, 0, -1, 0, -1, -1, -1, -2, -1, -1, -1, 0, -1, 0, -1, 0, 0, -1, -1, 1, -1, -1, -1, 0, -1, -2, ...].
a(n) is multiplicative with a(2^e) = -1 if e>0, a(3^e) = 0^e, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1 + (-1)^e)/2 if p == 5 (mod 6).
a(3*n) = a(6*n + 5) = 0.
a(2*n) = -A113448(n). a(6*n + 2) = -A033687(n).
a(3*n + 1) = A227696(n). a(6*n + 1) = A097195(n). a(12*n + 1) = A123884(n). a(12*n + 7) = 2 * A121361(n). - Michael Somos, Aug 03 2015

cp's OEIS Frontend

A112605 Number of representations of n as a sum of a square and six times a triangular number.

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A035178 a(n) = Sum_{d|n} Kronecker(-12, d) (= A134667(d)).

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A123484 Expansion of eta(q)^2 * eta(q^6)^4 * eta(q^8) * eta(q^24) / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.

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A122859 Expansion of phi(-q)^3 / phi(-q^3) in powers of q where phi() is a Ramanujan theta function.

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A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

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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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