A213624 -id:A213624 - cp's OEIS frontend

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

-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

A045831 Number of 4-core partitions of n.

Original entry on oeis.org

1, 1, 2, 3, 1, 3, 3, 3, 4, 4, 2, 2, 7, 3, 5, 6, 2, 4, 7, 3, 4, 7, 5, 8, 5, 4, 4, 8, 5, 6, 7, 2, 9, 11, 3, 8, 9, 4, 6, 5, 7, 5, 14, 7, 4, 10, 5, 10, 11, 3, 9, 10, 5, 8, 10, 4, 6, 15, 8, 9, 10, 6, 8, 15, 6, 10, 6, 5, 15, 9, 6, 8, 14, 8, 6, 13, 5, 16, 18, 7, 8, 7, 9, 6, 15, 6, 12, 17, 5, 8, 15, 7, 12

Offset: 0

N. J. A. Sloane

nonn

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Conjecturally  Sum_n a(n)q^(8n+5) equals theta series of sodalite. - Fred Lunnon, Mar 05 2015
Dickson writes that Liouville proved several related theorems about sums of triangular numbers. - Michael Somos, Feb 10 2020

			G.f. = 1 + x + 2*x^2 + 3*x^3 + x^4 + 3*x^5 + 3*x^6 + 3*x^7 + 4*x^8 + 4*x^9 + ...
G.f. = q^5 + q^13 + 2*q^21 + 3*q^29 + q^37 + 3*q^45 + 3*q^53 + 3*q^61 + 4*q^69 + ... ,
apparently the theta series of the sodalite net, aka edge-skeleton of space honeycomb by truncated octahedra. - _Fred Lunnon_, Mar 05 2015

L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. II, p. 23.

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from T. D. Noe)
M. Hirschhorn, and J. Sellers, Some amazing facts about 4-cores, J. Num. Thy. 60 (1996), 51-69.
K. Ono, and L. Sze, 4-core partitions and class numbers, Acta. Arith. 80 (1997), 249-272.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

A004024/4, column t=4 of A175595.
Cf. A286953.
Cf. A008443, A045818, A213624.

QP = QPochhammer; s = QP[q^4]^4/QP[q] + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Jul 26 2011, updated Nov 29 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^4 / eta(x + A), n))}; /* Michael Somos, Mar 24 2003 */

eta(32*z)^4/eta(8*z) = Sum_{x, y, z} q^(x^2+2*y^2+2*z^2), x, y, z >= 1 and odd.
From Michael Somos, Mar 24 2003: (Start)
Euler transform of period 4 sequence [1, 1, 1, -3, ...].
Expansion of q^(-5/8) * eta(q^4)^4/eta(q) in powers of q.
(End)
Number of solutions to n=t1+2*t2+2*t3 where t1, t2, t3 are triangular numbers. - Michael Somos, Jan 02 2006
G.f.: Product_{k>0} (1-q^(4*k))^4/(1-q^k).
Expansion of psi(q) * psi(q^2)^2 in powers of q where psi() is a Ramanujan theta function. - Michael Somos, Sep 02 2008

More terms from James Sellers, Feb 11 2000

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.

A033717 Number of integer solutions to the equation x^2 + 2y^2 + 4z^2 = n.

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 8, 8, 6, 6, 8, 4, 8, 12, 0, 8, 12, 8, 10, 12, 8, 8, 24, 8, 8, 14, 8, 16, 16, 4, 0, 16, 6, 16, 16, 8, 12, 20, 24, 8, 24, 8, 16, 20, 8, 20, 0, 16, 24, 18, 10, 8, 24, 12, 32, 24, 0, 16, 24, 12, 16, 20, 0, 24, 12, 8, 16, 28, 16, 16, 48, 8, 30, 32, 8, 20, 24, 16, 0, 16, 24, 18

Offset: 0

N. J. A. Sloane

nonn

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

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

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 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, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

A := Basis( ModularForms( Gamma1(16), 3/2), 82); A[1] + 2*A[2] + 2*A[3] + 4*A[4] + 4*A[5] + 4*A[6] + 8*A[7] + 8*A[8] + 6*A[9] + 8*A[10] + 4*A[11]; /* Michael Somos, Sep 03 2014 */

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^4], {q, 0, n}]; (* Michael Somos, Sep 03 2014 *)

{a(n) = my(G); if( n<0, 0, G = [1, 0, 0; 0, 2, 0; 0, 0, 4]; polcoeff( 1 + 2 * x * Ser(qfrep( G, n)), n))}; /* Michael Somos, Sep 03 2014 */

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^4 + A) * eta(x^8 + A)^3 / (eta(x + A)^2 * eta(x^16 + A)^2), n))}; /* Michael Somos, Sep 03 2014 */

Expansion of phi(q) * phi(q^2) * phi(q^4) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Sep 03 2014
Euler transform of period 16 sequence [2, -1, 2, -2, 2, -1, 2, -5, 2, -1, 2, -2, 2, -1, 2, -3, ...]. - Michael Somos, Sep 03 2014
G.f. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 8 (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Sep 03 2014
a(2*n + 1) = 2 * A045828(n). a(4*n) = A014455(n). a(4*n + 1) = 2 * A213625(n). a(4*n + 2) = 2 * A246811(n). a(4*n + 3) = 4 * A213624(n). - Michael Somos, Sep 03 2014
a(8*n) = A005875(n). a(8*n + 1) = 2 * A213622(n). a(8*n + 2) = 2 * A045834(n). a(8*n + 7) = 8 * A033763(n). - Michael Somos, Sep 03 2014
a(16*n) = A004015(n). a(16*n + 2) = 2 * A213022(n). a(16*n + 6) = 8 *
A008443(n). a(16*n + 8) = 2 * A045826(n). a(16*n + 10) = 8 * A045831(n). a(16*n + 14) = 0. - Michael Somos, Sep 03 2014
G.f.: theta_3(q) * theta_3(q^2) * theta_3(q^4).

A246833 Expansion of psi(-x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 1, -2, 3, -2, 4, -4, 2, -2, 5, -4, 2, -6, 3, -6, 7, -2, 5, -4, 5, -6, 6, -2, 5, -10, 3, -6, 10, -4, 6, -8, 3, -8, 7, -6, 7, -6, 4, -6, 11, -6, 9, -10, 3, -6, 14, -4, 8, -10, 8, -10, 5, -6, 4, -16, 7, -4, 10, -4, 13, -14, 7, -8, 8, -6, 10, -12, 7, -12

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 + x^2 - 2*x^3 + 3*x^4 - 2*x^5 + 4*x^6 - 4*x^7 + 2*x^8 - 2*x^9 + ...
G.f. = q^3 - 2*q^7 + q^11 - 2*q^15 + 3*q^19 - 2*q^23 + 4*q^27 - 4*q^31 + 2*q^35 + ...

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. A033763, A213624, A246815, A246832.

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

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

Expansion of q^(-3/4) * eta(q)^2 * eta(q^4) * eta(q^8)^2 / eta(q^2)^2 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 8 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246815.
a(n) = (-1)^n * A213624(n). a(2*n) = A246832(n). a(2*n + 1) = -2 * A033763(n).

A153182 Number of overpartitions of n with even M2-rank minus the number of overpartitions of n with odd M2-rank.

Original entry on oeis.org

1, 2, 4, 0, -2, 8, 8, -8, -4, 10, 8, 0, -8, 8, 16, -16, -10, 16, 12, 0, -8, 16, 8, -24, -8, 10, 24, 0, -16, 24, 16, -24, -20, 16, 16, 0, -10, 8, 24, -32, -8, 32, 16, 0, -24, 24, 16, -40, -24, 18, 28, 0, -8, 24, 32, -32, -16, 16, 8, 0, -32, 24, 32, -40, -26, 32

Offset: 0

Jeremy Lovejoy, Dec 20 2008

sign

a(8n+3) = 0.

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

G. C. Greubel, Table of n, a(n) for n = 0..1000
K. Bringmann and J. Lovejoy, Overpartitions and class numbers of binary quadratic forms

Cf. A005875, A045831, A213022, A213624, A213625, A256624.

a[n_]:= If[n < 0, 0, SeriesCoefficient[(1 + 8*Sum[(-1)^k x^(k^2 + 2*k)/(1 + x^(2*k))^2, {k, (Sqrt[4 n + 1] - 1)/2}])/EllipticTheta[4, 0, x], {x, 0, n}]]; Table[a[n], {n,0,30}] (* G. C. Greubel, Nov 29 2017 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 + 8 * sum(k=1, sqrtint(n+1)-1, (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2, A)) / (1 + 2 * sum(k=1, sqrtint(n), (-1)^k * x^k^2, A)), n))}; /* Michael Somos, Jul 13 2015 */

{a(n) = if( n<0, 0, polcoeff( 1 + 2 * sum(k=1, n, x^k / (1 + x^(2*k)) * prod(j=1, k,  (1 + x^(2*j - 1)) / (1 + x^(2*j)), 1 + x * O(x^(n-k)))), n))}; /* Michael Somos, Jul 13 2015 */

G.f.: 1 + 2Sum_{n >= 1} q^n(1+q)(1+q^3)...(1+q^(2n-1))/((1+q^2)(1+q^4)...(1+q^(2n))^2).
G.f.: (1 + 8 * Sum_{k>0} (-1)^k * x^(k^2 + 2*k) / (1 + x^(2*k))^2) / (1 + 2 * Sum_{k>0} (-1)^k * x^k^2). - Michael Somos, Jul 13 2015
a(4*n) = A256624(n) = 2 * a(n) - A005875(n). - Michael Somos, Jul 13 2015
a(8*n + 1) = 2 * A213022(n). a(8*n + 2) = 4 * A213625(n). a(8*n + 5) = 8 * A045831(n). a(8*n + 6) = 8 * A213624(n). - Michael Somos, Jul 13 2015

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).

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.

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

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A045831 Number of 4-core partitions of 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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A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^2 = n.

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A246833 Expansion of psi(-x)^2 * psi(x^4) in powers of x where psi() is a Ramanujan theta function.

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A153182 Number of overpartitions of n with even M2-rank minus the number of overpartitions of n with odd M2-rank.

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A033717 Number of integer solutions to the equation x^2 + 2y^2 + 4z^2 = n.

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