A131961 -id:A131961 - cp's OEIS frontend

A112604 Number of representations of n as a sum of three times a square and two times a triangular number.

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

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).

Number of representations of 2n as a sum of three times a triangular number and a triangular number.

			a(12) = 3 since we can write 12 = 3(2)^2 + 0 = 3(-2)^2 + 0 = 0 + 2*6.
2*12 = 24 = 3*1+21 = 3*3+15 = 3*6+6 so a(12) = 3.
G.f. = 1 + x^2 + 2*x^3 + 2*x^5 + x^6 + 2*x^9 + 3*x^12 + 2*x^14 + 2*x^15 + ... - _Michael Somos_, Aug 11 2009
G.f. = q + q^9 + 2*q^13 + 2*q^21 + q^25 + 2*q^37 + 3*q^49 + 2*q^57 + 2*q^61 + ... - _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

A112606(n) = a(2*n). 2 * A112607(n) = a(2*n + 1). A123884(n) = a(3*n). A112605(n) = a(3*n + 2). A131961(n) = a(6*n). A112608(n) =a(6*n + 2). 2 * A131963(n) = a(6*n + 3). 2 * A112609(n) = a(6*n + 5). - Michael Somos, Aug 11 2009
Cf. A002324, A033762, A093766, A164272.
Cf. A000700, A000122, A010054, A121373.

a[n_] := DivisorSum[4n+1, Switch[Mod[#, 3], 1, 1, 2, -1, 0, 0]&]; Table[ a[n], {n, 0, 104}] (* Jean-François Alcover, Dec 04 2015, adapted from PARI *)

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

{a(n) = my(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^6+A)^5 / eta(x^2+A)*(eta(x^4+A) / eta(x^3+A) / eta(x^12+A))^2, n))}; /* Michael Somos, Feb 14 2006 */

a(n) = A002324(4n+1) = A033762(2n) = d_{1, 3}(4n+1) - d_{2, 3}(4n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
From Michael Somos, Feb 14 2006: (Start)
Expansion of (psi(q)psi(q^3) + psi(-q)psi(-q^3))/2 in powers of q^2 where psi() is a Ramanujan theta function.
G.f.: (Sum_{k} x^k^2)^3*(Sum_{k>0} x^((k^2-k)/2))^2 = Product_{k>0} (1-x^(4k))(1-x^(6k))(1+x^(2k))(1+x^(3k))^2/(1+x^(6k))^2.
Euler transform of period 12 sequence [0, 1, 2, -1, 0, -2, 0, -1, 2, 1, 0, -2, ...]. (End)
From Michael Somos, Aug 11 2009: (Start)
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 A164272.
a(3*n + 1) = 0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 24 2023

A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 17 2006

nonn

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 + 3*x^4 + 2*x^5 + 2*x^6 + 2*x^8 + 2*x^9 + x^10 + ...
G.f. = q + 2*q^13 + q^25 + 2*q^37 + 3*q^49 + 2*q^61 + 2*q^73 + 2*q^97 + 2*q^109 + ...

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. A093829, A131961, A131962, A248886.

a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^2 EllipticTheta[ 4, 0, x^6] / EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^6] QPochhammer[ -x^2, x^2], {x, 0, n}]; (* Michael Somos, Oct 01 2015 *)

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

Expansion of q^(-1/12) * eta(q^2)^4 * eta(q^6)^2 / (eta(q)^2 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 2, -1, 2, -4, 2, -1, 2, -2, 2, -2, ...].
a(n) = A093829(12*n + 1).
a(n) = (-1)^n * A248886(n). a(2*n) = A131961(n). a(2*n + 1) = 2 * A131963(n). - Michael Somos, Oct 01 2015

A112606 Number of representations of n as a sum of six times a square and a triangular number.

Original entry on oeis.org

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

Offset: 0

James Sellers, Dec 21 2005

nonn

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

The greedy inverse starts 2, 0, 7, 6, 27, 300, 349, 14706, 216, 1035, 17107,... - R. J. Mathar, Apr 28 2020

			1 + x + x^3 + 3*x^6 + 2*x^7 + 2*x^9 + x^10 + 2*x^12 + x^15 + 2*x^16 + ...
q + q^9 + q^25 + 3*q^49 + 2*q^57 + 2*q^73 + q^81 + 2*q^97 + q^121 + 2*q^129 + ...
a(6) = 3 since we can write 6 = 6*1^2 + 0 = 6*(-1)^2 + 0 = 0 + 6.

M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.

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, A112608, A123484, A131961.

a[ n_] := If[ n < 0, 0, Sum[ KroneckerSymbol[ -3, d], {d, Divisors[ 8 n + 1]}]] (* Michael Somos, Jun 16 2011 since V6 *)
a[ n_] := If[ n < 0, 0, SeriesCoefficient[ EllipticTheta[ 3, 0, q^6] EllipticTheta[ 2, 0, q^(1/2)] / (2 q^(1/8)), {q, 0, n}]] (* Michael Somos, Jun 16 2011 *)

{a(n) = if( n<0, 0, n = 8*n + 1; sumdiv(n, d, kronecker(-3, d)))} /* Michael Somos, Sep 29 2006 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^12 + A)^5 / (eta(x + A) * eta(x^6 + A)^2 * eta(x^24 + A)^2), n))} /* Michael Somos, Sep 29 2006 */

a(n) = d_{1, 3}(8n+1) - d_{2, 3}(8n+1) where d_{a, m}(n) equals the number of divisors of n which are congruent to a mod m.
Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^12)^5 /(eta(q) * eta(q^6)^2 * eta(q^24)^2) in powers of q. - Michael Somos, Sep 29 2006
Expansion of phi(q^6) * psi(q) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 24 sequence [ 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -4, 1, -1, 1, -1, 1, 1, 1, -1, 1, -1, 1, -2, ...]. - Michael Somos, Sep 29 2006
G.f.: (Sum_{k} x^(6*k^2)) * (Sum_{k>0} x^((k^2-k)/2)). a(3*n+2)=0. - Michael Somos, Sep 29 2006
a(n) = A123484(24*n + 3) = A112604(2*n) = A112608(3*n). A131961(n) = a(3*n). A112608(n) = a(3*n + 1).

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

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

A137608 Expansion of (1 - psi(-q)^3 / psi(-q^3)) / 3 in powers of q where psi() is a Ramanujan theta function.

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, Jan 29 2008

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

Cf. A033762, A097195, A112604, A112605, A112606, A112607, A112608, A112609, A121361, A123884, A131961, A131962, A131963, A131964, A132973, A227696.

Cf. A035178, A093829, A113447 are same up to sign.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[n, KroneckerSymbol[ -12, #] &]]; (* Michael Somos, May 06 2015 *)
a[ n_] := SeriesCoefficient[ (4 + EllipticTheta[ 2, Pi/4, q^(1/2)]^3 / EllipticTheta[ 2, Pi/4, q^(3/2)]) / 6, {q, 0, n}]; (* Michael Somos, May 06 2015 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, {1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0}[[Mod[#, 12, 1]]] &]]; (* Michael Somos, May 07 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - 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))) / 3, n))}; /* Michael Somos, May 06 2015 */

Expansion of (1 - b(q^2)^2 / b(-q) ) / 3 in powers of q where b() is a cubic AGM function.
Moebius transform is period 12 sequence [ 1, -2, 0, 0, -1, 0, 1, 0, 0, 2, -1, 0, ...].
a(n) is multiplicative with a(2^e) = -1 unless 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} (-1)^k * (x^k + x^(3*k)) / (1 + x^k + x^(2*k)).
G.f.: ( 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 * A035178(n). -3 * a(n) = A132973(n) unless n = 0.
a(2*n) = -A035178(n). a(2*n + 1) = A033762(n). a(3*n) = a(n). a(3*n + 1) = A227696(n).
a(4*n + 1) + A112604(n). a(4*n + 3) = A112605(n). a(6*n + 1) = A097195(n). a(6*n + 5) = 0.
a(8*n + 1) = A112606(n). a(8*n + 3) = A112608(n). a(8*n + 5) = 2 * A112607(n-1). 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 * A131963(n). a(24*n + 19) = 2 * A131964(n).

A246650 Expansion of phi(x) * chi(-x) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 31 2014

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 - 3*x^5 - 2*x^6 + x^8 + 2*x^9 - 2*x^10 + ...
G.f. = q + q^4 - 2*q^7 + 2*q^13 - 3*q^16 - 2*q^19 + 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. A033687, A121361, A122861, A123884, A129451, A131961, A131962, A131963, A131964.

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

Expansion of q^(-1/3) * eta(q^2)^4 * eta(q^6)^2 / (eta(q) * eta(q^3) * eta(q^4)^2) in powers of q.
a(2*n) = A129451(n). a(4*n) = A123884(n). a(4*n + 1) = A122861(n). a(4*n + 2) = -2 * A121361(n). a(4*n + 3) = 0.
a(8*n) = A131961(n). a(8*n + 1) = A097195(n). a(8*n + 2) = -2 * A131962(n). a(8*n + 4) = 2 * A131963(n). a(8*n + 6) = -2 * A131964(n).
a(16*n + 1) = A123884(n). a(16*n + 5) = -3 * A033687(n). a(16*n + 9) = 2 * A121361(n). a(16*n + 13) = 0.

A248886 Expansion of f(-x, -x) * f(x^2, x^4) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Oct 01 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 + 3*x^4 - 2*x^5 + 2*x^6 + 2*x^8 - 2*x^9 + ...
G.f. = q - 2*q^13 + q^25 - 2*q^37 + 3*q^49 - 2*q^61 + 2*q^73 + 2*q^97 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A123884, A131961, A131963.

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

q='q+O('q^99); Vec(eta(q)^2*eta(q^4)*eta(q^6)^2/(eta(q^2)^2*eta(q^12))) \\ Altug Alkan, Jul 31 2018

Expansion of f(-x)^2 * phi(-x^6) / phi(-x^2) in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of phi(-x) * phi(-x^6) / chi(-x^2) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/12) * eta(q)^2 * eta(q^4) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [-2, 0, -2, -1, -2, -2, -2, -1, -2, 0, -2, -2, ...].
a(n) = (-1)^n * A123884(n). a(2*n) = A131961(n). a(2*n + 1) = -2 * A131963(n).

A260941 Expansion of phi(-x) * phi(x^6) / chi(-x^3) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Aug 04 2015

sign

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

			G.f. = 1 - 2*x + x^3 + 3*x^6 - 4*x^7 + 2*x^9 - 2*x^10 + 2*x^12 + x^15 + ...
G.f. = q - 2*q^9 + q^25 + 3*q^49 - 4*q^57 + 2*q^73 - 2*q^81 + 2*q^97 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A112608, A131961.

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^12 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) * eta(x^24 + A)^2), n))};

q='q+O('q^99); Vec(eta(q)^2*eta(q^12)^5/(eta(q^2)*eta(q^3)*eta(q^6)*eta(q^24)^2)) \\ Altug Alkan, Aug 01 2018

Expansion of q^(-1/8) * eta(q)^2 * eta(q^12)^5 / (eta(q^2) * eta(q^3) * eta(q^6) * eta(q^24)^2) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -4, -2, -1, -1, -1, -2, 1, -2, -1, -1, -1, -2, -2, ...].
a(3*n) = A131961(n). a(3*n + 1) = -2 * A112608(n). a(3*n + 2) = 0.

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A112604 Number of representations of n as a sum of three times a square and two times a triangular number.

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A123884 Expansion of phi(x) * phi(-x^6) / chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.

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A112606 Number of representations of n as a sum of six times a square and 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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A113447 Expansion of i * theta_2(i * q^3)^3 / (4 * theta_2(i * q)) in powers of q^2.

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