A192323 -id:A192323 - cp's OEIS frontend

A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q.

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0

Offset: 0

Michael Somos, Nov 14 2015

nonn

a(n) is the number of solutions in integers (x, y) of x^2 + 15*y^2 = n. - Michael Somos, Jul 17 2018

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^15 + 6*x^16 + 4*x^19 + 4*x^24 + 2*x^25 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500

Cf. A028625, A122855, A140727, A192323, A260675, A260676.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^15], {q, 0, n}];

{a(n) = if( n<1, n==0, qfrep([1, 0; 0, 15], n)[n]*2)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A))^2, n))};

q='q+O('q^99); Vec((eta(q^2)*eta(q^30))^5/(eta(q)*eta(q^4)*eta(q^15)*eta(q^60))^2) \\ Altug Alkan, Jul 18 2018

Expansion of (eta(q^2) * eta(q^30))^5 / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60))^2 in powers of q.
Euler transform of a period 60 sequence.
G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(15*k^2)).
a(3*n + 2) = a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
a(4*n) = A028625(n). a(4*n + 1) = 2 * A260675(n). a(4*n + 3) = 2 * A260676(n).
a(5*n) = A192323(n).
a(n) = A122855(n) + A140727(n).

A028956 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 4 ].

Original entry on oeis.org

1, 0, 4, 2, 0, 2, 0, 0, 8, 0, 0, 0, 6, 0, 0, 0, 0, 4, 4, 0, 6, 0, 0, 4, 0, 0, 0, 2, 0, 0, 4, 0, 12, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 2, 0, 4, 10, 0, 4, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 12, 0, 0, 0, 8, 0, 0, 2, 0, 0, 0, 0, 10, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 12, 4, 0, 4, 0, 0, 4, 0, 0, 0, 8

Offset: 0

N. J. A. Sloane

nonn

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

The number of integer solutions (x, y) to 2*x^2 + x*y + 2*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014

			G.f. =  1 + 4*q^4 + 2*q^6 + 2*q^10 + 8*q^16 + 6*q^24 + 4*q^34 + 4*q^36 + 6*q^40 + 4*q^46 + 2*q^54 + 4*q^60 + 12*q^64 + 8*q^76 + 2*q^90 + 4*q^94 + 10*q^96 + 4*q^100 + ...
G.f. = 1 + 4*x^2  + 2*x^3 + 2*x^5  + 8*x^8 + 6*x^12 + 4*x^17 + ... - _Michael Somos_, Jan 23 2023

R. Barman and N. D. Baruah, Theta function identities associated with Ramanujan's modular equations of degree 15, Proc. Indian Acad. Sci. Math. Sci. 120 (2010), no. 3, 267-284. see p. 271, equ. (3.1)

John Cannon, Table of n, a(n) for n = 0..5000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A192323.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^5], {q, 0, 4 n}]; (* Michael Somos, Aug 01 2011 *)

{a(n) = if( n<1, n==0, qfrep([4, 1;1, 4], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */

{a(n) = if( n<1, n==0, sumdiv( n, d, kronecker( -15, d) - kronecker( -3, d) * kronecker( 5, n/d)))}; /* Michael Somos, Aug 26 2006 */

Expansion of phi(q^3) * phi(q^5) + 4 * q^2 * psi(q^6) * psi(q^10) in powers of q where phi(),psi() are Ramanujan theta functions. - Michael Somos, Feb 09 2006
Expansion of (phi(q^3) * phi(q^5) + phi(-q^3) * phi(-q^5)) / 2 in powers of q^4 where phi() is a Ramanujan theta function. - Michael Somos, Aug 01 2011
Expansion of (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) - (eta(q) * eta(q^15))^2 / (eta(q^3) * eta(q^5)) in powers of q. - Michael Somos, Aug 26 2006
G.f.: theta_3(q^3) * theta_3(q^5) + theta_2(q^3) * theta_2(q^5) . - Michael Somos, Feb 09 2006
a(n) = A192323(4*n).

A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Offset: 1

Michael Somos, Jan 25 2017

nonn

			G.f. = x + 2*x^2 + 2*x^5 + x^6 + 2*x^7 + 4*x^10 + 2*x^15 + 2*x^17 + 2*x^22 + ...
G.f. = q^5 + 2*q^8 + 2*q^17 + q^20 + 2*q^23 + 4*q^32 + 2*q^47 + 2*q^53 + 2*q^65 + ...

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. A122855, A122856, A140727, A192323, A258276, A260649.

a[ n_] := If[ n < 1, 0, DivisorSum[ 3 n + 2, KroneckerSymbol[ -15, #] (-1)^Boole[Mod[#, 4] == 2] &]];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^30] QPochhammer[ -x^25, x^30] QPochhammer[ x^30], {x, 0, n}];

{a(n) = if( n<1, 0, sumdiv(3*n + 2, d, kronecker(-15, d) * (-1)^(d%4==2) ))};

{a(n) = if( n<1, 0, my(m = 3*n + 2, s, x); for(y=1, sqrtint(m\5), if( y%3 && issquare((m - 5*y^2)\3, &x), s += (x>0) + 1)); s)};

{a(n) = if( n<1, 0, my(A); n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^10 + A)^2 * eta(x^15 + A) * eta(x^60 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^5 + A) * eta(x^20 + A) * eta(x^30 + A)), n))};

G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(15*k^2 - 10*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(30*k-25)) * (1 + x^(30*k-5)) * (1 - x^(30*k)).
a(n) = A122855(3*n + 2) = A260649(3*n + 2) = A122856(6*n + 4) = A258276(6*n + 4).
a(n) = - A140727(3*n + 2). 2 * a(n) = A192323(3*n + 2).

A317642 Expansion of theta_3(q^2)*theta_3(q^5), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation 2*x^2 + 5*y^2 = n.

			G.f. = 1 + 2*q^2 + 2*q^5 + 4*q^7 + 2*q^8 + 4*q^13 + 2*q^18 + 2*q^20 + 4*q^22 + ...

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000286, A020674, A033718, A106889, A108563, A192323.

nmax = 98; CoefficientList[Series[EllipticTheta[3, 0, q^2] EllipticTheta[3, 0, q^5], {q, 0, nmax}], q]
nmax = 98; CoefficientList[Series[QPochhammer[-q^2, -q^2] QPochhammer[-q^5, -q^5]/(QPochhammer[q^2, -q^2] QPochhammer[q^5, -q^5]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(4*k-2))^2*(1 - x^(4*k))*(1 + x^(10*k-5))^2*(1 - x^(10*k)).

A317643 Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

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

Offset: 0

Ilya Gutkovskiy, Aug 02 2018

nonn

Number of integer solutions to the equation 3*x^2 + 4*y^2 = n.

			G.f. = 1 + 2*q^3 + 2*q^4 + 4*q^7 + 2*q^12 + 6*q^16 + 4*q^19 + 2*q^27 + 4*q^28 + ...

N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Cf. A000049, A020677, A068229, A108563, A192323.

nmax = 100; CoefficientList[Series[EllipticTheta[3, 0, q^3] EllipticTheta[3, 0, q^4], {q, 0, nmax}], q]
nmax = 100; CoefficientList[Series[QPochhammer[-q^3, -q^3] QPochhammer[-q^4, -q^4]/(QPochhammer[q^3, -q^3] QPochhammer[q^4, -q^4]), {q, 0, nmax}], q]

G.f.: Product_{k>=1} (1 + x^(6*k-3))^2*(1 - x^(6*k))*(1 + x^(8*k-4))^2*(1 - x^(8*k)).

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A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q.

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A028956 Theta series of quadratic form (or lattice) with Gram matrix [ 4, 1; 1, 4 ].

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A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.

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A317642 Expansion of theta_3(q^2)*theta_3(q^5), where theta_3() is the Jacobi theta function.

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A317643 Expansion of theta_3(q^3)*theta_3(q^4), where theta_3() is the Jacobi theta function.

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