A045828 -id:A045828 - cp's OEIS frontend

A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

1, -2, -2, 4, 3, -2, -6, 4, 4, -6, -4, 4, 7, -8, -2, 8, 8, -4, -10, 4, 4, -10, -10, 8, 9, -4, -6, 12, 8, -6, -10, 12, 4, -14, -8, 4, 16, -10, -8, 8, 9, -10, -12, 12, 8, -12, -12, 4, 20, -10, -6, 20, 8, -6, -10, 12, 8, -20, -18, 8, 11, -12, -12, 16, 8, -6, -20

Offset: 0

Michael Somos, Sep 03 2014

sign

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

			G.f. = 1 - 2*x - 2*x^2 + 4*x^3 + 3*x^4 - 2*x^5 - 6*x^6 + 4*x^7 + 4*x^8 + ...
G.f. = q - 2*q^3 - 2*q^5 + 4*q^7 + 3*q^9 - 2*q^11 - 6*q^13 + 4*q^15 + ...

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. A033763, A045828, A213622.

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

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

Expansion of q^(-1/2) * eta(q)^2 * eta(q^2) * eta(q^8)^2 / eta(q^4)^2 in powers of q
Euler transform of period 8 sequence [ -2, -3, -2, -1, -2, -3, -2, -3, ...].
a(n) = (-1)^floor((n+1) / 2) * A045828(n).
a(4*n) = A213622(n). a(4*n + 3) = 4 * A033763(n).

A246836 Expansion of phi(x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -2, -4, 3, 2, -6, -4, 4, 6, -4, -4, 7, 8, -2, -8, 8, 4, -10, -4, 4, 10, -10, -8, 9, 4, -6, -12, 8, 6, -10, -12, 4, 14, -8, -4, 16, 10, -8, -8, 9, 10, -12, -12, 8, 12, -12, -4, 20, 10, -6, -20, 8, 6, -10, -12, 8, 20, -18, -8, 11, 12, -12, -16, 8, 6, -20

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 + 2*x - 2*x^2 - 4*x^3 + 3*x^4 + 2*x^5 - 6*x^6 - 4*x^7 + 4*x^8 + ...
G.f. = q + 2*q^3 - 2*q^5 - 4*q^7 + 3*q^9 + 2*q^11 - 6*q^13 - 4*q^15 + 4*q^17 + ...

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. A045828, A246833, A246835.

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

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^4), n))};

Expansion of q^(-1/2) * eta(q^2)^7 * eta(q^8)^2 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 8 sequence [ 2, -5, 2, -1, 2, -5, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246835.
a(n) = (-1)^floor(n/2) * A045828(n). a(n) = (-1)^n * A246815(n).
a(2*n) = A246835(n). a(2*n + 1) = 2 * A246833(n).

A004025 Theta series of b.c.c. lattice with respect to long edge.

Original entry on oeis.org

2, 4, 0, 0, 8, 8, 0, 0, 10, 8, 0, 0, 8, 16, 0, 0, 16, 12, 0, 0, 16, 8, 0, 0, 10, 24, 0, 0, 24, 16, 0, 0, 16, 16, 0, 0, 8, 24, 0, 0, 32, 16, 0, 0, 24, 16, 0, 0, 18, 28, 0, 0, 24, 32, 0, 0, 16, 8, 0, 0, 24, 32, 0, 0, 32, 32, 0, 0, 32, 16, 0, 0, 16, 40, 0, 0, 32

Offset: 1

N. J. A. Sloane

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

The body-centered cubic (b.c.c. also known as D3*) lattice is the set of all triples [a, b, c] where the entries are all integers or all one half an odd integer. A long edge is centered at a triple with two integer entries and the remaining entry is one half an odd integer. - Michael Somos, May 31 2012

			2*q + 4*q^2 + 8*q^5 + 8*q^6 + 10*q^9 + 8*q^10 + 8*q^13 + 16*q^14 + 16*q^17 + ...

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 1..1000
N. J. A. Sloane and B. K. Teo, Theta series and magic numbers for close-packed spherical clusters, J. Chem. Phys. 83 (1985) 6520-6534.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to b.c.c. lattice

Cf. A045828, A045834, A045836.

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[2*QPochhammer[x^2+A]^5 * (QPochhammer[x^8+A]^4 / (QPochhammer[x+A]^2*QPochhammer[x^4+A]^4)), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from PARI *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( 2 * eta(x^2 + A)^5 * eta(x^8 + A)^4 / (eta(x + A)^2 * eta(x^4 + A)^4), n))} /* Michael Somos, May 31 2012 */

From Michael Somos, May 31 2012: (Start)
Expansion of 2 * x * phi(x) * psi(x^4)^2 = 2 * x * psi(-x^2)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of 2 * eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.
a(4*n) = a(4*n + 3) = 0. a(n) = 2 * A045836(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n). (End)

A246631 Number of integer solutions to x^2 + 2y^2 + 2z^2 = n.

Original entry on oeis.org

1, 2, 4, 8, 6, 8, 8, 0, 12, 10, 8, 24, 8, 8, 16, 0, 6, 16, 12, 24, 24, 16, 8, 0, 24, 10, 24, 32, 0, 24, 16, 0, 12, 16, 16, 48, 30, 8, 24, 0, 24, 32, 16, 24, 24, 24, 16, 0, 8, 18, 28, 48, 24, 24, 32, 0, 48, 16, 8, 72, 0, 24, 32, 0, 6, 32, 32, 24, 48, 32, 16, 0

Offset: 0

Michael Somos, Aug 31 2014

nonn

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

			G.f. = 1 + 2*q + 4*q^2 + 8*q^3 + 6*q^4 + 8*q^5 + 8*q^6 + 12*q^8 + 10*q^9 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
Mariia Dospolova, Ekaterina Kochetkova, and Eric T. Mortenson, A new Andrews--Crandall-type identity and the number of integer solutions to x^2+2y^2+2z^2=n, arXiv:2307.05244 [math.NT], 2023.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A005875, A008443, A014455, A033717, A045826, A045828, A045831, A045834, A212885, A213022, A213624, A213625, A246811.

A := Basis( ModularForms( Gamma0(8), 3/2), 80); A[1] + 2*A[2];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^2, {q, 0, n}];

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

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

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 0; 0, 0, 2 ].
Expansion of phi(q) * phi(q^2)^2 = phi(-q^4)^4 / phi(-q) in powers of q where phi() is a Ramanujan theta function.
Expansion of eta(q^2) * eta(q^4)^8 / (eta(q)^2 * eta(q^8)^4) in powers of q.
Euler transform of period 8 sequence [ 2, 1, 2, -7, 2, 1, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 4 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A014455.
G.f.: theta_3(q) * theta_3(q^2)^2.
G.f.: Product{k>0} (1 - x^(2*k)) * (1 - x^(4*k))^8 / ((1 - x^k)^2 * (1 - x^(8*k))^4).
G.f.: Product{k>0} (1 + x^(2*k)) * (1 + x^k)^2 * (1 - x^(4*k))^3 / (1 + x^(4*k))^4.
a(n) = (-1)^floor((n+1) / 2) * A212885(n) = abs(A212885(n)).
a(n) = A033717(2*n). a(2*n) = A014455(n). a(2*n + 1) = 2 * A246811(n).
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = 4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 3) = 8 * A008443(n). a(8*n + 4) = 2 * A045826(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). a(8*n + 7) = 0.

A246953 Expansion of phi(-x) * psi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 2, -4, 3, -2, 6, -4, 4, -6, 4, -4, 7, -8, 2, -8, 8, -4, 10, -4, 4, -10, 10, -8, 9, -4, 6, -12, 8, -6, 10, -12, 4, -14, 8, -4, 16, -10, 8, -8, 9, -10, 12, -12, 8, -12, 12, -4, 20, -10, 6, -20, 8, -6, 10, -12, 8, -20, 18, -8, 11, -12, 12, -16, 8, -6, 20

Offset: 0

Michael Somos, Sep 08 2014

sign

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

			G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 3*x^4 - 2*x^5 + 6*x^6 - 4*x^7 + 4*x^8 + ...
G.f. = q - 2*q^2 + 2*q^3 - 4*q^4 + 3*q^5 - 2*q^6 + 6*q^7 - 4*q^8 + 4*q^9 + ...

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. A045828, A213624, A213625, A246954.

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

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

Expansion of psi(x^2) * psi(-x)^2 = psi(-x)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q)^2 * eta(q^4)^4 / eta(q^2)^3 in powers of q.
Euler transform of period 4 sequence [ -2, 1, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 128^(1/2) * (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A246954.
G.f.: Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4.
a(n) = (-1)^n * A045828(n). a(2*n) = A213625(n). a(2*n + 1) = - 2 * A213624(n).

A286399 Expansion of eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 in powers of q.

Original entry on oeis.org

0, 0, 1, 8, 32, 96, 244, 528, 1024, 1856, 3126, 5016, 7808, 11616, 16808, 23856, 32768, 44352, 59293, 77352, 100032, 128128, 161052, 201264, 249856, 305280, 371294, 450128, 537856, 640992, 762744, 894528, 1048576, 1228224, 1419858, 1642080, 1897376, 2167008

Offset: 0

Seiichi Manyama, May 08 2017

nonn

The average order of a(n) is n^5 * Pi^6 / 30720. - Vaclav Kotesovec, Feb 09 2023

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

Cf. A013973 (E_6), A045828.

CoefficientList[x^2 * Series[QPochhammer[x^2]^12 * QPochhammer[x^4]^8 / QPochhammer[x]^8, {x, 0, 40}], x] (* Vaclav Kotesovec, Feb 08 2023 *)

G.f.: x^2 * Product_{k>0} (1 - x^(2 * k))^12 * (1 - x^(4 * k))^8 / (1 - x^k)^8.

A290705 Theta series of triamond.

Original entry on oeis.org

1, 3, 0, 6, 0, 6, 8, 12, 6, 9, 0, 6, 0, 18, 0, 12, 12, 12, 0, 18, 0, 12, 24, 12, 8, 21, 0, 24, 0, 6, 0, 24, 6, 24, 0, 12, 0, 30, 24, 12, 24, 12, 0, 30, 0, 30, 0, 24, 24, 27, 0, 12, 0, 18, 32, 36, 0, 24, 0, 18, 0, 30, 0, 36, 12, 12, 0, 42, 0, 24, 48, 12, 30

Offset: 0

Andrey Zabolotskiy, Aug 09 2017

nonn

Theta series with respect to a node of a lattice known as triamond, Laves graph [embedded in space], K_4 lattice, (10,3)-a, or srs net. This lattice possesses the "strong isotropic" property; the only other lattice that has this property in 3 dimensions is the diamond lattice. Unlike diamond, triamond is chiral.

A004013 and 3*A045828, interleaved.

Toshikazu Sunada, Crystals that nature might miss creating, Notices Amer. Math. Soc. 55 (No. 2, 2008), 208-215.
Toshikazu Sunada, Correction to "Crystals That Nature Might Miss Creating", Notices Amer. Math. Soc., 55 (No. 3, 2008), page 343. [Annotated scanned copy]
Wikipedia, Laves graph

Cf. A004013, A005925, A045828, A113062.

See A038620 for coordination sequence.

(* count lattice sites straightforwardly *)
cell = Join @@ ({#, # + {1, 1, 1}/2} & /@ {{0, 0, 0}, {1/4, 0, 1/4}, {-1/4, -1/4, 0}, {0, 1/4, -1/4}}); (* lattice sites in a conventional bcc unit cell *)
n = 10;
s = O[q]^(n^2 + 1) + Sum[q^(8 Norm[a + {i, j, k}]^2), {i, -n-1, n+1}, {j, -n-1,  n+1}, {k, -n-1, n+1}, {a, cell}];
CoefficientList[Normal[s], q] &
(* or use the generation function *)
a[n_] := SeriesCoefficient[ EllipticTheta[3, 0, x^8]^3 + EllipticTheta[ 2, 0, x^8]^3 + 3/4 EllipticTheta[3, 0, x^2] EllipticTheta[2, 0, x^2]^2, {x, 0, n}];

A045836 Half of theta series of b.c.c. lattice with respect to long edge.

Original entry on oeis.org

1, 2, 0, 0, 4, 4, 0, 0, 5, 4, 0, 0, 4, 8, 0, 0, 8, 6, 0, 0, 8, 4, 0, 0, 5, 12, 0, 0, 12, 8, 0, 0, 8, 8, 0, 0, 4, 12, 0, 0, 16, 8, 0, 0, 12, 8, 0, 0, 9, 14, 0, 0, 12, 16, 0, 0, 8, 4, 0, 0, 12, 16, 0, 0, 16, 16, 0, 0, 16, 8, 0, 0, 8, 20, 0, 0, 16, 8, 0, 0, 17

Offset: 1

N. J. A. Sloane

nonn

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

The body-centered cubic (b.c.c. also known as D3*) lattice is the set of all triples [a, b, c] where the entries are all integers or all one half an odd integer. A long edge is centered at a triple with two integer entries and the remaining entry is one half an odd integer. - Michael Somos, May 31 2012

			q + 2*q^2 + 4*q^5 + 4*q^6 + 5*q^9 + 4*q^10 + 4*q^13 + 8*q^14 + 8*q^17 + ...

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

Cf. A004025, A045828, A045834.

a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[QPochhammer[x^2+A]^5 * (QPochhammer[x^8+A]^4 / (QPochhammer[x+A]^2*QPochhammer[x^4+A]^4)), {x, 0, n}]]; Table[a[n], {n, 0, 80}] (* Jean-François Alcover, Nov 05 2015, adapted from PARI *)

{a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A)^4 / (eta(x + A)^2 * eta(x^4 + A)^4), n))} /* Michael Somos, May 31 2012 */

From Michael Somos, May 31 2012: (Start)
Expansion of x * phi(x) * psi(x^4)^2 = x * psi(-x^2)^4 / phi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q^2)^5 * eta(q^8)^4 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Euler transform of period 8 sequence [ 2, -3, 2, 1, 2, -3, 2, -3, ...].
a(4*n) = a(4*n + 3) = 0. a(n) = A004025(n) / 2. a(4*n + 1) = A045834(n). a(4*n + 2) = 2 * A045828(n). (End)

A319078 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, -4, -8, 6, 8, -8, 0, 12, 10, -8, -24, 8, 8, -16, 0, 6, 16, -12, -24, 24, 16, -8, 0, 24, 10, -24, -32, 0, 24, -16, 0, 12, 16, -16, -48, 30, 8, -24, 0, 24, 32, -16, -24, 24, 24, -16, 0, 8, 18, -28, -48, 24, 24, -32, 0, 48, 16, -8, -72, 0, 24, -32, 0, 6, 32

Offset: 0

Michael Somos, Sep 09 2018

sign

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

			G.f. = 1 + 2*x - 4*x^2 - 8*x^3 + 6*x^4 + 8*x^5 - 8*x^6 + 12*x^8 + ...

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

Cf. A004015, A004024, A005887, A008443, A045834, A083703, A080965, A132429, A212885, A213624.

A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];

a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 3, 0, q]^2, {q, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 4, 0, q^2]^2, {q, 0, n}];

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

Expansion of eta(q^2)^9 / (eta(q)^2 * eta(q^4)^4) in powers of q.
Expansion of phi(q) * phi(-q^2)^2 = phi(-q^2)^4 / phi(-q) in powers of q.
Euler transform of period 4 sequence [2, -7, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2^(11/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045834.
G.f. Product_{k>0}  (1 - x^k)^3 * (1 + x^k)^5 / (1 + x^(2*k))^4.
a(n) = (-1)^n * A212885(n) = A083703(2*n) = A080965(2*n).
a(4*n) = a(n) * -A132429(n + 2) where A132429 is a period 4 sequence.
a(4*n) = A005875(n). a(4*n + 1) = 2 * A045834(n). a(4*n + 2) = -4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = -4 * A213625(n). a(8*n + 3) = -8 * A008443(n). a(8*n + 4) = A005887(n). a(8*n + 5) = 2 * A004024(n). a(8*n + 6) = -8 * A213624(n). a(8*n + 7) = 0.

cp's OEIS Frontend

A246815 Expansion of phi(-x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A246836 Expansion of phi(x) * psi(-x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A004025 Theta series of b.c.c. lattice with respect to long edge.

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A246631 Number of integer solutions to x^2 + 2*y^2 + 2*z^2 = n.

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A246953 Expansion of phi(-x) * psi(x^2)^2 in powers of x where phi(), psi() are Ramanujan theta functions.

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A286399 Expansion of eta(q^2)^12 * eta(q^4)^8 / eta(q)^8 in powers of q.

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A290705 Theta series of triamond.

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A246631 Number of integer solutions to x^2 + 2y^2 + 2z^2 = n.