A260671 -id:A260671 - cp's OEIS frontend

A192323 Expansion of theta_3(q^3) * theta_3(q^5) in powers of q.

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

Offset: 0

Michael Somos, Aug 01 2011

nonn

			G.f. = 1 + 2*q^3 + 2*q^5 + 4*q^8 + 2*q^12 + 4*q^17 + 2*q^20 + 4*q^23 + 2*q^27 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A028956, A122855, A140727, A260671.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^5], {q, 0, n}];

{a(n) = if( n<1, n==0, qfrep([3, 0; 0, 5], n)[n]*2)};

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^5 / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A))^2, n))};

Expansion of (eta(q^6) * eta(q^10))^5 / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20))^2 in powers of q.
Euler transform of a period 60 sequence.
G.f.: (Sum_{k} x^(3 * k^2)) * (Sum_{k} x^(5 * k^2)).
a(3*n + 1) = a(4*n + 2) = a(5*n + 1) = a(5*n + 4) = 0. a(4*n) = A028956(n).
a(n) = A122855(n) - A140727(n). a(5*n) = A260671(n).

A028625 Expansion of (theta_3(z)theta_3(15z)+theta_2(z)theta_2(15z)).

Original entry on oeis.org

1, 2, 0, 0, 6, 0, 4, 0, 0, 2, 4, 0, 0, 0, 0, 2, 10, 0, 0, 4, 0, 0, 0, 0, 8, 2, 0, 0, 0, 0, 0, 4, 0, 0, 8, 0, 6, 0, 0, 0, 8, 0, 0, 0, 0, 0, 8, 0, 0, 2, 0, 4, 0, 0, 4, 0, 0, 0, 0, 0, 6, 4, 0, 0, 14, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 12, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 4, 0, 0, 0, 8, 0, 12, 0, 0, 0, 6, 0, 0, 0

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Theta series of quadratic form (or lattice) with Gram matrix [ 2, 1; 1, 8 ].
The number of integer solutions (x, y) to x^2 + x*y + 4*y^2 = n, discriminant -15. - Ray Chandler, Jul 12 2014
a(n) = number of solutions in integers (x, y) of x^2 + 15*y^2 = 4*n. - Michael Somos, Jul 17 2018

			G.f. = 1 + 2*q^2 + 6*q^8 + 4*q^12 + 2*q^18 + 4*q^20 + 2*q^30 + 10*q^32 + 4*q^38 + 8*q^48 + 2*q^50 + 4*q^62 + 8*q^68 + 6*q^72 + 8*q^80 + 8*q^92 + 2*q^98 + ...
G.f. = 1 + 2*x + 6*x^4 + 4*x^6 + 2*x^9 + 4*x^10 + 2*x^15 + 10*x^16 + 4*x^19 + ... - _Michael Somos_, Jul 17 2018

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. A260671.

r[n_] := Reduce[x^2 + x*y + 4*y^2 == n, {x, y}, Integers]; Table[rn = r[n]; Which[rn === False, 0, Head[rn] === Or, Length[rn], Head[rn] === And, 1], {n, 0, 105}] (* Jean-François Alcover, Nov 05 2015, after the comment by Ray Chandler *)
a[0] = 1; a[n_] := With[{K = KroneckerSymbol}, DivisorSum[n, K[-15, #] + K[#, 3]*K[n/#, 5]&]]; Table[a[n], {n, 0, 103}] (* Jean-François Alcover, Jul 07 2017, after Michael Somos *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 3, 0, x^15] + EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^15], {x, 0, n}]; (* Michael Somos, Jul 17 2018 *)

{a(n) = if( n<1, n==0, qfrep([2, 1;1, 8], n, 1)[n]*2)}; /* Michael Somos, Aug 26 2006 */

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

Expansion of phi(q) * phi(q^15) + 4 * q^4 * psi(q^2) * psi(q^30) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 26 2006
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
a(n) = A260671(4*n). - Michael Somos, Jul 17 2018

A260675 Expansion of psi(x^2) * phi(x^15) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 14 2015

nonn

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

			G.f. = 1 + x^2 + x^6 + x^12 + 2*x^15 + 2*x^17 + x^20 + 2*x^21 + 2*x^27 + ...
G.f. = q + q^9 + q^25 + q^49 + 2*q^61 + 2*q^69 + q^81 + 2*q^85 + 2*q^109 + ...

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. A260671.

a[ n_] := SeriesCoefficient[ (1/2) x^(-1/4) EllipticTheta[ 2, 0, x] EllipticTheta[ 3, 0, x^15], {x, 0, n}];

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

Expansion of q^(-1/4) * eta(q^4)^2 * eta(q^30)^5 / (eta(q^2) * eta(q^15)^2 * eta(q^60)^2) in powers of q.
Euler transform of a period 60 sequence.
2 * a(n) = A260671(4*n + 1).

A260676 Expansion of phi(x) * psi(x^30) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Nov 14 2015

nonn

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

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + x^30 + 2*x^31 + 2*x^34 + ...
G.f. = q^15 + 2*q^19 + 2*q^31 + 2*q^51 + 2*q^79 + 2*q^115 + q^135 + 2*q^139 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A260671.

m:=130;
f:= func< q | (&*[( (1-q^(2*n))^5*(1-q^(60*n))^2 )/( (1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n)) ): n in [1..m+1]]) >;
R:=PowerSeriesRing(Integers(), m);
Coefficients(R!( f(x) )); // G. C. Greubel, Feb 02 2023

a[ n_] := SeriesCoefficient[ EllipticTheta[3, 0, x] QPochhammer[x^60]^2 / QPochhammer[x^30], {x, 0, n}];

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

m = 130
def f(q): return product( ((1-q^(2*n))^5*(1-q^(60*n))^2)/((1-q^n)^2*(1-q^(4*n))^2*(1-q^(30*n))) for n in range(1,m+2))
def A260676_list(prec):
    P. = PowerSeriesRing(ZZ, prec)
    return P( f(x) ).list()
A260676_list(m) # G. C. Greubel, Feb 02 2023

Expansion of q^(-15/4) * eta(q^2)^5 * eta(q^60)^2 / (eta(q)^2 * eta(q^4)^2 * eta(q^30)) in powers of q.
Euler transform of a period 60 sequence.
2 * a(n) = A260671(4*n + 15).

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A192323 Expansion of theta_3(q^3) * theta_3(q^5) in powers of q.

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A028625 Expansion of (theta_3(z)*theta_3(15z)+theta_2(z)*theta_2(15z)).

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A260675 Expansion of psi(x^2) * phi(x^15) in powers of x where phi(), psi() are Ramanujan theta functions.

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A260676 Expansion of phi(x) * psi(x^30) in powers of x where phi(), chi() are Ramanujan theta functions.

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A028625 Expansion of (theta_3(z)theta_3(15z)+theta_2(z)theta_2(15z)).