A033715 -id:A033715 - cp's OEIS frontend

A318984 Theta series of quadratic form x^2 + xy + 17y^2.

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

Offset: 0

Jianing Song, Sep 06 2018

Number of integer solutions (x, y) to x^2 + x*y + 17*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-67)].

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 4*x^17 + 4*x^19 + 4*x^23 + 2*x^25 + 4*x^29 + 2*x^36 + 4*x^37 + 4*x^47 + 2*x^49 + 4*x^59 + 2*x^64 + 2*x^67 + 4*x^68 + 4*x^71 + 4*x^73 + 4*x^76 + ...

Jianing Song, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318982.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), this sequence (d=-67), A318985 (d=-163).

Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-67, #] &]];
2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *)

a(n) = if(n, 2*sumdiv(n, d, kronecker(-67, d)), 1)

G.f.: 1 + 2 * Sum_{k>0} Kronecker(-67, k) * x^k / (1 - x^k).
a(n) = 2 * A318982(n) unless n = 0.
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(67^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-67, p) = -1, b(p^e) = e + 1 if Kronecker(-67, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(67) = 0.767613... . - Amiram Eldar, Dec 16 2023

A318985 Theta series of quadratic form x^2 + xy + 41y^2.

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

Offset: 0

Jianing Song, Sep 06 2018

Number of integer solutions (x, y) to x^2 + x*y + 41*y^2 = n. Also, a(n) is the number of integral elements with norm n in Q[sqrt(-163)].

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^16 + 2*x^25 + 2*x^36 + 4*x^41 + 4*x^43 + 4*x^47 + 2*x^49 + 4*x^53 + 4*x^61 + 2*x^64 + 4*x^71 + ...

Jianing Song, Table of n, a(n) for n = 0..10000
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS.

Cf. A318983.

Number of integer solutions to f(x,y) = n where f(x,y) is the principal binary quadratic form with discriminant d: A004016 (d=-3), A004018 (d=-4), A002652 (d=-7), A033715 (d=-8), A028609 (d=-11), A028641 (d=-19), A138811 (d=-43), A318984 (d=-67), this sequence (d=-163).

Join[{1}, a[n_]:=If[n<0, 0, DivisorSum[n, KroneckerSymbol[-163, #] &]];
2 Table[a[n], {n, 1, 110}]] (* Vincenzo Librandi, Sep 10 2018 *)

a(n) = if(n, 2*sumdiv(n, d, kronecker(-163, d)), 1)

G.f.: 1 + 2 * Sum_{k>0} Kronecker(-163, k) * x^k / (1 - x^k).
a(n) = 2 * A318983(n) unless n = 0.
a(0) = 1, a(n) = 2 * b(n) for n > 0, where b() is multiplicative with b(163^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if Kronecker(-163, p) = -1, b(p^e) = e + 1 if Kronecker(-163, p) = 1.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = 2*Pi/sqrt(163) = 0.49213705... . - Amiram Eldar, Dec 16 2023

A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, Apr 08 2008

sign

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

a(n) is nonzero if and only if n is in A002479.

			G.f. = 1 + 2*q - 2*q^2 - 4*q^3 + 2*q^4 - 4*q^6 + 2*q^8 + 6*q^9 - 4*q^11 + ...

G. C. Greubel, Table of n, a(n) for n = 0..10000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A002479, A033715, A033761, A082564, A112603, A125095, A133692.

A := Basis( ModularForms( Gamma1(32), 1), 105); A[1] + 2*A[2] - 2*A[3] - 4*A[4] + 2*A[5] - 4*A[7] + 2*A[9] + 6*A[10] - 4*A[12] + 4*A[13] + 4*A[16]; /* Michael Somos, Aug 29 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 29 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^7 / (QPochhammer[ q]^2 QPochhammer[ q^4]^3), {q, 0, n}]; (* Michael Somos, Feb 18 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], 2 (-1)^Quotient[n, 2] Sum[ JacobiSymbol[ -2, d], {d, Divisors @ n}]]; (* Michael Somos, Feb 18 2015 *)

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

{a(n) = my(A); if ( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 / (eta(x + A)^2 * eta(x^4 + A)^3), n))};

Expansion of eta(q^2)^7 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -5, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 8^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A112603.
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k-1))^2 / (1 + x^(2*k)).
a(8*n + 5) = a(8*n + 7) = 0.
a(n) = (-1)^n * A082564(n). a(2*n) = A133692(n). a(2*n + 1) = 2 * A125095(n). a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = 2 * A112603(n). a(8*n + 3) = -4 * A033761(n).

A320246 Expansion of Product_{k=1..12} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 200, 252, 360, 456, 564, 770, 940, 1178, 1532, 1852, 2256, 2858, 3430, 4100, 5086, 5982, 7076, 8612, 10040, 11672, 13960, 16068, 18496, 21866, 24796, 28288, 32924, 37074, 41876, 48156, 53732, 60014, 68546, 75836, 83996

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_12) to the equation a_1^2 + 2*a_2^2 + ... + 12*a_12^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8), A320241 (m=9), A320242 (m=10), this sequence (m=12), A320247 (m=16).

Cf. A320067.

A320247 Expansion of Product_{k=1..16} theta_3(q^k), where theta_3() is the Jacobi theta function.

Original entry on oeis.org

1, 2, 2, 6, 8, 10, 22, 26, 36, 60, 78, 106, 152, 202, 258, 370, 478, 600, 822, 1032, 1310, 1720, 2140, 2656, 3418, 4222, 5172, 6510, 7922, 9636, 11928, 14424, 17268, 21088, 25236, 29996, 36222, 42824, 50544, 60252, 70830, 82832, 97732, 113956, 132242, 154866, 179164, 206396

Offset: 0

Seiichi Manyama, Oct 08 2018

nonn

Also the number of integer solutions (a_1, a_2, ... , a_16) to the equation a_1^2 + 2*a_2^2 + ... + 16*a_16^2 = n.

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Eric Weisstein's World of Mathematics, Jacobi Theta Functions

Product_{k=1..m} theta_3(q^k): A000122 (m=1), A033715 (m=2), A029594 (m=3), A320139 (m=4), A320231 (m=5), A320232 (m=6), A320233 (m=7), A320234 (m=8), A320241 (m=9), A320242 (m=10), A320246 (m=12), this sequence (m=16).

Cf. A320067.

A156384 The number of solutions to x^2 + y^2 + 2*z^2 = n in nonnegative integers x,y,z.

Original entry on oeis.org

1, 2, 2, 2, 3, 2, 2, 2, 2, 4, 4, 2, 4, 4, 0, 2, 3, 4, 6, 4, 4, 2, 4, 2, 2, 6, 4, 6, 6, 2, 0, 4, 2, 6, 8, 2, 7, 6, 4, 2, 4, 4, 6, 6, 4, 6, 0, 4, 4, 6, 6, 4, 10, 4, 6, 6, 0, 6, 10, 4, 6, 6, 0, 6, 3, 4, 8, 8, 8, 4, 6, 2, 6, 10, 4, 6, 10, 4, 0, 4, 4, 8, 14, 6, 6, 8, 4, 6, 4, 6, 10, 6, 6, 6, 0, 2, 2, 12, 8, 8

Offset: 0

R. H. Hardin Feb 09 2009

nonn

Also, the number of 4X4 matrices composed of squares of integers, symmetric under 90 degree rotation, with all rows summing to n. Such matrices have the form:
z^2 x^2 y^2 z^2
y^2 z^2 z^2 x^2
x^2 z^2 z^2 y^2
z^2 y^2 x^2 z^2
with x^2 + y^2 + 2*z^2 = n.

			All matrices for n=9:
...0.0.9.0...0.9.0.0...4.0.1.4...4.1.0.4
...9.0.0.0...0.0.0.9...1.4.4.0...0.4.4.1
...0.0.0.9...9.0.0.0...0.4.4.1...1.4.4.0
...0.9.0.0...0.0.9.0...4.1.0.4...4.0.1.4

Cf. A213024

a(n) = ( A014455(n) + 2*A033715(n) + A004018(n) + A000122(n/2) + 2*A000122(n) + A000007(n) )/8. - Max Alekseyev, Sep 29 2012

G.f.: (1 + theta_3(q))^2*(1 + theta_3(q^2))/8, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

More general definition from Max Alekseyev, Sep 29 2012

A213024 The number of solutions to x^2 + y^2 + 2*z^2 = n in positive integers x,y,z.

Original entry on oeis.org

0, 0, 0, 0, 1, 0, 0, 2, 0, 0, 2, 0, 2, 2, 0, 2, 1, 0, 2, 2, 2, 2, 2, 2, 0, 2, 2, 2, 6, 0, 0, 4, 0, 2, 4, 2, 3, 4, 2, 2, 2, 0, 6, 4, 2, 4, 0, 4, 2, 4, 2, 0, 8, 2, 2, 6, 0, 2, 8, 2, 6, 4, 0, 6, 1, 0, 4, 6, 4, 4, 6, 2, 2, 6, 2, 4, 8, 4, 0, 4, 2, 2, 10, 4, 6, 4, 2, 6, 2, 2, 8, 6, 6, 6, 0, 2, 0, 8, 6, 2, 9

Offset: 0

Max Alekseyev, Sep 29 2012

nonn

Cf. A156384

N=166; x='x+O('x^N);
T(x)=sum(k=1, 1+sqrtint(N), x^(k*k) );
gf=T(x)^2 * T(x^2);
v=Vec('a0 + gf );  v[1]=0;  v
/* Joerg Arndt, Oct 01 2012 */

a(n) = ( A014455(n) - 2*A033715(n) - A004018(n) + A000122(n/2) + 2*A000122(n) - A000007(n) )/8.

G.f.: T(x)^2 * T(x^2) where T(x) = sum(k>=1, x^(k^2)). [Joerg Arndt, Oct 01 2012]

A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 31 2013

nonn

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

			G.f. = 1 + 2*q + 2*q^4 + 2*q^8 + 6*q^9 + 4*q^12 + 2*q^16 + 4*q^17 + 4*q^24 + 2*q^25 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..2500 from G. C. Greubel)
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A033715, A112603, A113411, A242609.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^16 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^8 + A) * eta(x^32 + A))^2, n))};

Expansion of (eta(q^2) * eta(q^16))^5 / (eta(q) * eta(q^4) * eta(q^8) * eta(q^32))^2 in powers of q.
Euler transform of period 32 sequence [2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -4, 2, -3, 2, -1, 2, -3, 2, 1, 2, -3, 2, -1, 2, -3, 2, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 32^(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^8)^k^2).
a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.  a(4*n) = a(8*n) = A033715(n). a(4*n + 1) = A033715(4*n + 1). a(8*n + 1) = 2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
(-1)^n * a(n) = A242609(n). - Michael Somos, Feb 20 2015

A242609 Expansion of phi(-q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Offset: 0

Michael Somos, May 19 2014

sign

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

			G.f. = 1 - 2*q + 2*q^4 + 2*q^8 - 6*q^9 + 4*q^12 + 2*q^16 - 4*q^17 + ...

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. A033715, A112603, A113411, A226225.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q^8], {q, 0, n}];

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

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^16 + A)^5 / (eta(x^2 + A) * eta(x^8 + A)^2 * eta(x^32 + A)^2), n))};

Expansion of eta(q)^2 * eta(q^16)^5 / (eta(q^2) * eta(q^8)^2 * eta(q^32)^2) in powers of q.
G.f.: (Sum_{k in Z} (-x)^k^2) * (Sum_{k in Z} (x^8)^k^2).
a(4*n + 2) = a(4*n + 3) = a(8*n + 5) = 0.  a(4*n) = a(8*n) = A033715(n). a(8*n + 1) = -2 * A112603(n). a(8*n + 4) = 2 * A113411(n).
a(n) = (-1)^n * A226225(n).

A244540 Expansion of phi(q) * (phi(q) + phi(q^2)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 3, 3, 2, 3, 4, 2, 0, 3, 5, 4, 2, 2, 4, 0, 0, 3, 6, 5, 2, 4, 0, 2, 0, 2, 7, 4, 4, 0, 4, 0, 0, 3, 4, 6, 0, 5, 4, 2, 0, 4, 6, 0, 2, 2, 4, 0, 0, 2, 3, 7, 4, 4, 4, 4, 0, 0, 4, 4, 2, 0, 4, 0, 0, 3, 8, 4, 2, 6, 0, 0, 0, 5, 6, 4, 2, 2, 0, 0, 0, 4, 7, 6, 2, 0, 8, 2

Offset: 0

Michael Somos, Jun 29 2014

nonn

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

			G.f. = 1 + 3*q + 3*q^2 + 2*q^3 + 3*q^4 + 4*q^5 + 2*q^6 + 3*q^8 + 5*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A000122, A000700, A004018, A010054, A033715, A033761, A053692, A093709, A107635, A121373, A244526, A244543.

A := Basis( ModularForms( Gamma1(8), 1), 33); A[1] + 3*A[2] + 3*A[3];

a[ n_] := If[ n < 1, Boole[n == 0], Sum[ {3, 0, -1, 0, 1, 0, -3, 0}[[ Mod[ d, 8, 1] ]], {d, Divisors @ n}]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] (EllipticTheta[ 3, 0, q] + EllipticTheta[ 3, 0, q^2]) / 2, {q, 0, n}];

{a(n) = if( n<1, n==0, sumdiv(n, d, [0, 3, 0, -1, 0, 1, 0, -3][d%8 + 1]))};

{a(n) = my(A); if( n<0, 0, A = sum(k=1, sqrtint(n), 2 * x^k^2, 1 + x * O(x^n)); polcoeff( A * (A + subst(A, x, x^2)) / 2, n))};

A = ModularForms( Gamma1(8), 1, prec=33) . basis(); A[0] + 3*A[1] + 3*A[2];

Expansion of f(-q^3, -q^5)^2 * phi(q) / psi(-q) = f(-q^3, -q^5)^2 * chi(q)^3 in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [3, -3, 1, 0, 1, -3, 3, -2, ...].
Moebius transform is period 8 sequence [3, 0, -1, 0, 1, 0, -3, 0, ...].
Convolution product of A244526 and A107635. Convolution product of A000122 and A093709.
a(n) = (A004018(n) + A033715(n)) / 2 = A244543(2*n).
a(2*n) = a(n). a(8*n + 3) = 2*A033761(n). a(8*n + 5) = 4*A053692(n). a(8*n + 7) = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=0..m} a(k) = Pi*(1 + 1/sqrt(2))/2 = 2.681517... . - Amiram Eldar, Jun 08 2025

cp's OEIS Frontend

A318984 Theta series of quadratic form x^2 + x*y + 17*y^2.

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A318985 Theta series of quadratic form x^2 + x*y + 41*y^2.

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A139093 Expansion of phi(q) * phi(-q^2) in powers of q where phi() is a Ramanujan theta function.

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A320246 Expansion of Product_{k=1..12} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A320247 Expansion of Product_{k=1..16} theta_3(q^k), where theta_3() is the Jacobi theta function.

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A156384 The number of solutions to x^2 + y^2 + 2*z^2 = n in nonnegative integers x,y,z.

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A213024 The number of solutions to x^2 + y^2 + 2*z^2 = n in positive integers x,y,z.

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PARI

Formula

A226225 Expansion of phi(q) * phi(q^8) in powers of q where phi() is a Ramanujan theta function.

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Mathematica

A318984 Theta series of quadratic form x^2 + xy + 17y^2.

A318985 Theta series of quadratic form x^2 + xy + 41y^2.