cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-7 of 7 results.

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

Views

Author

Michael Somos, Jul 05 2015

Keywords

Comments

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

Examples

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

Links

Crossrefs

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.

Programs

  • Mathematica
    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)};
  • PARI
    {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)))};

Formula

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

A033717 Number of integer solutions to the equation x^2 + 2*y^2 + 4*z^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

Views

Author

N. J. A. Sloane

Keywords

Comments

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

Examples

			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 + ...

References

  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p 102 eq 9.

Links

Crossrefs

Cf. A004015, A008443, A014455, A033763, A045826, A045828, A045831, A045834, A213022, A213622, A213624, A213625, A246811.

Programs

  • Magma
    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 */
  • Mathematica
    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 *)
  • PARI
    {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 */
  • PARI
    {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 */

Formula

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).

A246835 Expansion of psi(-x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -6, 4, -4, 7, -2, 8, -10, 4, -10, 9, -6, 8, -10, 4, -8, 16, -8, 9, -12, 8, -12, 20, -6, 8, -10, 8, -18, 11, -12, 8, -20, 12, -8, 20, -6, 20, -26, 8, -8, 15, -10, 16, -18, 12, -16, 20, -10, 16, -16, 8, -24, 24, -8, 21, -26, 8, -20, 20, -14, 8, -28
Offset: 0

Views

Author

Michael Somos, Sep 04 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 3*x^2 - 6*x^3 + 4*x^4 - 4*x^5 + 7*x^6 - 2*x^7 + 8*x^8 + ...
G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...

Links

Crossrefs

Cf. A132969, A213622, A213625, A246836.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(-1/4) * eta(q)^2 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.
a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(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

Views

Author

Jeremy Lovejoy, Dec 20 2008

Keywords

Comments

a(8n+3) = 0.

Examples

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

Links

Crossrefs

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

Programs

  • Mathematica
    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 *)
  • PARI
    {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 */
  • PARI
    {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 */

Formula

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 + 2*y^2 + 2*z^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

Views

Author

Michael Somos, Aug 31 2014

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

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

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(8), 3/2), 80); A[1] + 2*A[2];
  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2]^2, {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0, 0; 0, 2, 0; 0, 0, 2], n)[n])};
  • PARI
    {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))};

Formula

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

Views

Author

Michael Somos, Sep 08 2014

Keywords

Comments

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

Examples

			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 + ...

Links

Crossrefs

Cf. A045828, A213624, A213625, A246954.

Programs

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

Formula

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

Views

Author

Michael Somos, Sep 09 2018

Keywords

Comments

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

Examples

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

Links

Crossrefs

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

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(16), 3/2), 66); A[1] + 2*A[2] - 4*A[3] - 8*A[4];
  • Mathematica
    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}];
  • PARI
    {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))};

Formula

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.
Showing 1-7 of 7 results.

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