A005877 -id:A005877 - cp's OEIS frontend

A045828 One fourth of theta series of cubic lattice with respect to face.

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, 16, 12, 14, 8, 12, 20, 14, 6, 12, 20, 8, 26, 12, 8, 22, 8, 12, 15

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number of solutions to n = t1 + t2 + 2*t3 where  t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
The cubic lattice is the set of triples [a, b, c] where the entries are all integers. A face is centered at a triple where one entry is an integer and the other two are one half an odd integer. - Michael Somos, Jun 29 2012

			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 + 6*x^9 + ...
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 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.

Seiichi Manyama, 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. A005877, A005884, A033761, A010054, A033763, A212885.

a[ n_] := SeriesCoefficient[ 1/4 EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x]^2, {x, 0, n + 1/2}]; (* Michael Somos, Jun 29 2012 *)
a[ n_] := SeriesCoefficient[ 1/8 EllipticTheta[ 2, 0, x^2] EllipticTheta[ 2, 0, x]^2, {x, 0, 2 n + 1}]; (* Michael Somos, Jun 29 2012 *)
QP = QPochhammer; s = (QP[q^2]^3*QP[q^4]^2)/QP[q]^2 + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A)^2 / eta(x + A)^2, n))}; /* Michael Somos, Oct 25 2006 */

Expansion of q^(-1/2) * (eta(q^2)^3 * eta(q^4)^2) / eta(q)^2 in powers of q. - Michael Somos, Jan 02 2006
Expansion of phi(x) * psi(x^2)^2 = psi(x)^2 * psi(x^2) = psi(x)^4 / phi(x) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Jun 29 2012
Euler transform of period 4 sequence [2, -1, 2, -3, ...]. - Michael Somos, Mar 05 2003
Convolution of A033761 and A010054. - Michael Somos, Jun 29 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = (1/2)^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A212885. - Michael Somos, Sep 08 2018

Edited by Michael Somos, Mar 05 2003

A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

Original entry on oeis.org

-1, 0, 0, 4, 6, 0, 0, 12, 12, 0, 0, 12, 16, 0, 0, 24, 18, 0, 0, 12, 24, 0, 0, 36, 24, 0, 0, 16, 24, 0, 0, 36, 36, 0, 0, 24, 30, 0, 0, 48, 24, 0, 0, 12, 48, 0, 0, 60, 40, 0, 0, 24, 24, 0, 0, 48, 48, 0, 0, 36, 48, 0, 0, 60, 42, 0, 0, 12, 48, 0, 0, 84, 36, 0, 0

Offset: 0

Michael Somos, Jul 05 2015

sign

Coefficients of q-expansion of Eisenstein series G_{3/2}(tau) multiplied by 12. - N. J. A. Sloane, Mar 16 2019

			G.f. = -1 + 4*x^3 + 6*x^4 + 12*x^7 + 12*x^8 + 12*x^11 + 16*x^12 + 24*x^15 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Kathrin Bringmann and Jeremy Lovejoy, Overpartitions and class numbers of binary quadratic forms, arXiv:0712.0631 [math.NT], 2007. See page 5, equation (1.12).
Archer Clayton, Helen Dai, Tianyu Ni, Erick Ross, Hui Xue, and Jake Zummo, Non-repetition of second coefficients of Hecke polynomials, arXiv:2411.18419 [math.NT], 2024. See p. 21.
Wikipedia, Hurwitz class number.
Don Zagier, Modular Forms of One Variable, Notes based on a course given in Utrecht, 1991. See page 50.

Cf. A004024, A005869, A005876, A005877, A005878, A005884, A005886, A008443.
Cf. A045828, A045831, A045834, A058305, A058306, A130695, A181648, A185220.
Cf. A188569, A213022, A213617, A213624, A213625, A213627, A259827.
See also A306934-A306937.

terms = 100; gf[m_] := With[{r = Range[-m, m]}, -2 Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, r}]/EllipticTheta[3, 0, x] - 2 Sum[(-1)^k*x^(k^2 + 2 k)/(1 + x^(2 k))^2, {k, r}]/EllipticTheta[3, 0, -x]]; gf[terms // Sqrt // Ceiling] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Apr 02 2017 *)
a[ n_] := If[ n<1, -Boole[n==0], With[{m = Floor[(-1 + Sqrt[1 + 4*n])/2]}, -2*SeriesCoefficient[ Sum[(-1)^k*x^(k^2 + k)/(1 + (-x)^k)^2, {k, -m-1,m}] / EllipticTheta[3, 0, x] + Sum[(-1)^k*x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, -m-2,m}]/ EllipticTheta[3, 0, -x], {x, 0, n}]]]; (* Michael Somos, Feb 04 2022 *)

PARI
```
{a(n) = 12 * qfbhclassno(n)};
```

{a(n) = my(D, f); 12 * if( n<1, (n==0)/-12, [D, f] = core(-n, 1); if( D%4>1 && !(f%2), D*=4; f/=2); if( D%4<2, qfbclassno(D) / max(1, D+6), 0) * sumdiv(f, d, moebius(d) * kronecker(D, d) * sigma(f/d)))};

a(n) = 12 * A058305(n) / A058306(n). a(4*n + 1) = a(4*n + 2) = 0. a(3*n + 4) = 6 * A259827(n).
a(4*n + 3) = 4 * A130695(n). a(8*n + 3) = A005886(n) = 2 * A005869(n) = 4 * A008443(n). a(12*n + 7) = 12 * A259655(n).
a(16*n + 4) = 6 * A045834(n) = 3 * A005876(n). a(16*n + 8) = 12 * A045828(n) = 6 * A005884(n) = 3 * A005877(n).
a(24*n + 3) = 4 * A213627(n). a(24*n + 7) = 12 * A185220(n). a(24*n + 11) = 12 * A213617(n). a(24*n + 19) = 12 * A181648(n). a(24*n + 23) = 12 * A188569(n+1).
a(32*n + 4) = 6 * A213022(n). a(32*n + 8) = 12 * A213625(n). a(32*n + 12) = 16 * A008443(n) = 8 * A005869(n) = 4 * A005886(n) = 2 * A005878(n). a(32*n + 20) = 24 * A045831(n) = 6 * A004024(n). a(32*n + 24) = 24 * A213624(n).
G.f.: -2 * (Sum_{k in Z} (-1)^k * x^(k*k + k) / (1 + (-x)^k)^2) / (Sum_{k in Z} x^k^2) - 2 * (Sum_{k in Z} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (Sum_{k in Z} (-x)^k^2).
a(n) >= 0 if n > 0. - Michael Somos, Feb 04 2022

A014455 Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. Number of integer solutions to x^2 + y^2 + 2*z^2 = n.

Original entry on oeis.org

1, 4, 6, 8, 12, 8, 8, 16, 6, 12, 24, 8, 24, 24, 0, 16, 12, 16, 30, 24, 24, 16, 24, 16, 8, 28, 24, 32, 48, 8, 0, 32, 6, 32, 48, 16, 36, 40, 24, 16, 24, 16, 48, 40, 24, 40, 0, 32, 24, 36, 30, 16, 72, 24, 32, 48, 0, 32, 72, 24, 48, 40, 0, 48, 12, 16, 48, 56, 48, 32, 48, 16, 30, 64

Offset: 0

N. J. A. Sloane

This is the tetragonal P lattice (the classical holotype) of dimension 3.

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

			G.f. = 1 + 4*q + 6*q^2 + 8*q^3 + 12*q^4 + 8*q^5 + 8*q^6 + 16*q^7 + 6*q^8 + 12*q^9 + ...

John Cannon, Table of n, a(n) for n = 0..10000
G. Nebe and N. J. A. Sloane, Home page for this lattice
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A005875, A005877, A008443, A033763, A045826, A045828, A156384, A213024, A246631.

A := Basis( ModularForms( Gamma1(8), 3/2), 40); A[1] + 4*A[2] + 6*A[3] + 8*A[4]; /* Michael Somos, Aug 31 2014 */

r[n_, z_] := Reduce[x^2 + y^2 + 2*z^2 == n, {x, y}, Integers]; a[n_] := Module[{rn0, rnz, k0, k}, rn0 = r[n, 0]; k0 = If[rn0 === False, 0, If[Head[rn0] === And, 1, Length[rn0]]]; For[k = 0; z = 1, z <= Ceiling[Sqrt[n/2]], z++, rnz = r[n, z]; If[rnz =!= False, k = If[Head[rnz] === And, k+1, k + Length[rnz]]]]; k0 + 2*k]; Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Oct 07 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2 EllipticTheta[ 3, 0, q^2], {q, 0, n}]; (* Michael Somos, Aug 31 2014 *)
QP = QPochhammer; s = QP[q^2]^8*(QP[q^4]/(QP[q]^4*QP[q^8]^2)) + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015, after Michael Somos *)

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 1, 0; 0, 0, 2], n)[n])}; /* Michael Somos, Jul 05 2005 */

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^8 * eta(x^4 + A) / (eta(x + A)^4 * eta(x^8 + A)^2), n))}; /* Michael Somos, Jul 05 2005 */

Expansion of phi(q)^2 * phi(q^2) = psi(q)^4 / psi(q^4) in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Apr 07 2012
Expansion of eta(q^2)^8 * eta(q^4) / (eta(q)^4 * eta(q^8)^2) in powers of q. - Michael Somos, Jul 05 2005
Euler transform of period 8 sequence [4, -4, 4, -5, 4, -4, 4, -3, ...]. - Michael Somos, Jul 07 2005
G.f.: theta_3(q)^2 * theta_3(q^2) = Product_{k>0} (1 - x^(2*k))^8 * (1 - x^(4*k)) / ((1 - x^k)^4 * (1 - x^(8*k))^2).
There is a classical formula (essentially due to Gauss): Write (uniquely) -2n=D(2^vf)^2, with D<0 fundamental discriminant, f odd, v>=-1. Then a(n)=12L((D/.),0)(1-(D/2))\sum_{d\mid f}\mu(d)(D/d)sigma(f/d) (the formula for A005875), except that the factor (1-(D/2)) has to be replaced by 1/3 if v=-1 and by 1 if v=0 (and kept if v>=1). Here mu() is the Moebius function, (D/2) and (D/d) are Kronecker-Legendre symbols, sigma() is the sum of divisors function, L((D/.),0)=h(D)/(w(D)/2) is the value at 0 of the L() function of the quadratic character (D/.), equal to the class number h(D) divided by 2 or 3 in the special cases D=-4 and -3. - Henri Cohen (Henri.Cohen(AT)math.u-bordeaux1.fr), May 12 2010
a(2*n) = a(8*n) = A005875(n). a(2*n + 1) = A005877(n) = 4 * A045828(n). a(4*n) = A004015(n). a(4*n + 2) = 2 * A045826(n). a(8*n + 4) = 12 * A045828(n). a(8*n + 7) = 16 * A033763(n). a(16*n + 6) = 8 * A008443(n). a(16*n + 14) = 0. - Michael Somos, Apr 07 2012
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246631.

A212885 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

Offset: 0

Michael Somos, May 29 2012

sign

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..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004015, A004024, A005875, A005877, A005878, A005887, A008443, A045828, A045834, A213624.

a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q]* EllipticTheta[3, 0, -q]^2, {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 30 2017 *)

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

Expansion of phi(-x) * phi(-x^2)^2 = phi(-x^2)^4 / phi(x) in powers of x where phi() is a Ramanujan theta function.
Expansion of eta(q^2)^3 * eta(q)^2 / eta(q^4)^2 in powers of q.
Euler transform of period 4 sequence [-2, -5, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 32 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A045828.
G.f.: Product_{k>0} (1 - x^(2*k))^3 * (1 - x^k)^2 / (1 - x^(4*k))^2.
a(4*n) = A005875(n). a(4*n + 1) = -2 * A045834(n). a(4*n + 2) = - A005877(n) = -4 * A045828(n).
a(8*n) = A004015(n). a(8*n + 3) = A005878(n) = 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

A045828 One fourth of theta series of cubic lattice with respect to face.

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A259825 a(n) = 12*H(n) where H() is the Hurwitz class number.

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A014455 Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 1, 0; 0, 0, 2 ]. Number of integer solutions to x^2 + y^2 + 2*z^2 = n.

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A212885 Expansion of phi(q) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.

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