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 1-3 of 3 results.

A033762 Product t2(q^d); d | 3, where t2 = theta2(q) / (2 * q^(1/4)).

Original entry on oeis.org

1, 1, 0, 2, 1, 0, 2, 0, 0, 2, 2, 0, 1, 1, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 3, 0, 0, 0, 2, 0, 2, 2, 0, 2, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 0, 4, 2, 0, 2, 0, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 1, 0, 0, 2, 2, 0, 4, 0, 0, 2, 0, 0, 0, 3, 0, 2, 0, 0, 2, 0, 0, 2, 0, 0, 3, 2, 0
Offset: 0

Views

Author

N. J. A. Sloane

Keywords

Comments

Number of solutions of 8*n + 4 = x^2 + 3*y^2 in positive odd integers. - Michael Somos, Sep 18 2004
Half the number of integer solutions of 4*n + 2 = x^2 + y^2 + z^2 where 0 = x + y + z and x and y are odd. - Michael Somos, Jul 03 2011
Given g.f. A(x), then q^(1/2) * 2 * A(q) is denoted phi_1(z) where q = exp(Pi i z) in Conway and Sloane.
Half of theta series of planar hexagonal lattice (A2) with respect to an edge.
Bisection of A002324. Number of ways of writing n as a sum of a triangular plus three times a triangular number [Hirschhorn]. - R. J. Mathar, Mar 23 2011
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + x + 2*x^3 + x^4 + 2*x^6 + 2*x^9 + 2*x^10 + x^12 + x^13 + 2*x^15 + ...
G.f. = q + q^3 + 2*q^7 + q^9 + 2*q^13 + 2*q^19 + 2*q^21 + q^25 + q^27 + 2*q^31 + ...
a(6) = 2 since 8*6 + 4 = 52 = 5^2 + 3*3^2 = 7^2 + 3*1^2.

References

  • Burce C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, 1991, see p. 223 Entry 3(i).
  • J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, 1999, p. 103. See Eq. (13).
  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.27).

Links

Crossrefs

Cf. A002324, A004016, A005881, A035178, A091393, A093766, A093829, A096936, A112298, A113447, A113661, A113974, A115979, A122860, A123331, A123484, A136748, A137608.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 1), 202); A[2] + A[4]; /* Michael Somos, Jul 25 2014 */
  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, Mod[(3 - #)/2, 3, -1] &]]; (* Michael Somos, Jul 03 2011 *)
QP = QPochhammer; s = (QP[q^2]*QP[q^6])^2/(QP[q]*QP[q^3]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
a[ n_] := If[ n < 1, Boole[n == 0], Times @@ (Which[# < 2, 0^#2, Mod[#, 6] == 5, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger@(2 n + 1))]; (* Michael Somos, Mar 06 2016 *)
%t A033762 a[ n_] := SeriesCoefficient[ (1/4) x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)], {x, 0, n}]; (* Michael Somos, Mar 06 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A)), n))}; /* Michael Somos, Sep 18 2004 */
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, kronecker( -12, d) * (n / d % 2)))}; /* Michael Somos, Nov 04 2005 */
  • PARI
    {a(n) = if( n<0, 0, n = 8*n + 4; sum( j=1, sqrtint( n\3), (j%2) * issquare(n - 3*j^2)))} /* Michael Somos, Nov 04 2005 */
  • PARI
    {a(n) = if( n<0, 0, sumdiv(2*n + 1, d, kronecker(-3, d)))}; /* Michael Somos, Mar 06 2016 */

Formula

Expansion of q^(-1/2) * (eta(q^2) * eta(q^6))^2 / (eta(q) * eta(q^3)) in powers of q. - Michael Somos, Apr 18 2004
Expansion of q^(-1) * (a(q) - a(q^4)) / 6 in powers of q^2 where a() is a cubic AGM theta function. - Michael Somos, Oct 24 2006
Expansion of psi(x) * psi(x^3) in powers of x where psi() is a Ramanujan theta function. - Michael Somos, Jul 03 2011
Euler transform of period 6 sequence [ 1, -1, 2, -1, 1, -2, ...]. - Michael Somos, Apr 18 2004
From Michael Somos, Sep 18 2004: (Start)
Given g.f. A(x), then B(x) = (x * A(x^2))^2 satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = v^3 + 4*u*v*w + 16*v*w^2 - 8*w*v^2 - w*u^2.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = (1 + (-1)^e) / 2 if p==5 (mod 6) otherwise b(p^e) = e+1. (Clarification: the g.f. A(x) is not the primary function of interest, but rather B(x) = x * A(x^2), which is an eta-quotient and is the generating function of a multiplicative sequence.)
G.f.: (Sum_{j>0} x^((j^2 - j) / 2)) * (Sum_{k>0} x^(3(k^2 - k) / 2)) = Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(3*k)) * (1 - x^(6*k)).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k * (1 - x^k) * (1 - x^(4*k)) * (1 - x^(5*k)) / (1 - x^(12*k)). (End)
G.f.: s(4)^2*s(12)^2/(s(2)*s(6)), where s(k) := subs(q=q^k, eta(q)), where eta(q) is Dedekind's function, cf. A010815. [Fine]
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} x^k / (1 + x^k + x^(2*k)) - x^(4*k) / (1 + x^(4*k) + x^(8*k)). - Michael Somos, Nov 04 2005
a(n) = A002324(2*n + 1) = A035178(2*n + 1) = A091393(2*n + 1) = A093829(2*n + 1) = A096936(2*n + 1) = A112298(2*n + 1) = A113447(2*n + 1) = A113661(2*n + 1) = A113974(2*n + 1) = A115979(2*n + 1) = A122860(2*n + 1) = A123331(2*n + 1) = A123484(2*n + 1) = A136748(2*n + 1) = A137608(2*n + 1). A005881(n) = 2*a(n).
6 * a(n) = A004016(6*n + 3). - Michael Somos, Mar 06 2016
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(3)) = 0.906899... (A093766). - Amiram Eldar, Nov 23 2023

Extensions

Corrected by Charles R Greathouse IV, Sep 02 2009

A113973 Expansion of phi(x^3)^3/phi(x) where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 4, -2, 2, 0, 4, -4, 4, -2, 0, 0, 2, -4, 8, 0, 2, 0, 4, -4, 0, -4, 0, 0, 4, -2, 8, -2, 4, 0, 0, -4, 4, 0, 0, 0, 2, -4, 8, -4, 0, 0, 8, -4, 0, 0, 0, 0, 2, -6, 4, 0, 4, 0, 4, 0, 8, -4, 0, 0, 0, -4, 8, -4, 2, 0, 0, -4, 0, 0, 0, 0, 4, -4, 8, -2, 4, 0, 8, -4, 0, -2, 0, 0, 4, 0, 8, 0, 0, 0, 0, -8, 0, -4, 0, 0, 4, -4, 12, 0, 2, 0, 0, -4, 8
Offset: 0

Views

Author

Michael Somos, Nov 10 2005

Keywords

Comments

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

References

  • Bruce C. Berndt, Ramanujan's Notebooks Part V, Springer-Verlag, 1985, see p. 375, Entry 35.

Links

Crossrefs

a(n)=-2*A113974(n) if n>0.
Cf. A000700, A000122, A010054, A073010, A121373.

Programs

  • Mathematica
    s = EllipticTheta[3, 0, q^3]^3/EllipticTheta[3, 0, q] + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 04 2015 *)
f[p_, e_] := If[Mod[p, 6] == 1, e + 1, (1 + (-1)^e)/2]; f[2, e_] := ((-1)^e - 3)/2; f[3, e_] := 1; a[0] = 1; a[1] = -2; a[n_] := -2 * Times @@ f @@@ FactorInteger[n]; Array[a, 100, 0] (* Amiram Eldar, Nov 14 2023 *)
  • PARI
    {a(n)=local(x); if(n<1, n==0, x=valuation(n,2); if(n%2,-2,(3-(-1)^x))*sumdiv(n/2^x,d, kronecker(-3,d)))}
  • PARI
    {a(n)=local(A,p,e); if(n<1, n==0, A=factor(n); -2*prod(k=1,matsize(A)[1], if(p=A[k,1], e=A[k,2]; if(p==2, (-3+(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
  • PARI
    {a(n)=if(n<1, n==0, -2*direuler(p=2,n, if(p==2, 2-(1+2*X)/(1-X^2), 1/(1-X)/(1-kronecker(-3,p)*X)))[n])}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=sum(k=1,sqrtint(n), 2*x^k^2, 1+x*O(x^n)); polcoeff( subst(A+x*O(x^(n\3)),x,x^3)^3/A, n))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)^2*eta(x^4+A)^2*eta(x^6+A)^15/ eta(x^2+A)^5/eta(x^3+A)^6/eta(x^12+A)^6, n))}

Formula

a(n) = -2*b(n) where b(n) is multiplicative and b(2^e) = (1-3(-1)^e)/2 if e>0, b(3^e) = 1, b(p^e) = e+1 if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
Euler transform of period 12 sequence [ -2, 3, 4, 1, -2, -6, -2, 1, 4, 3, -2, -2, ...].
Moebius transform is period 12 sequence [ -2, 6, 0, -2, 2, 0, -2, 2, 0, -6, 2, 0, ...].
Expansion of (eta(q)^2*eta(q^4)^2*eta(q^6)^15)/ (eta(q^2)^5*eta(q^3)^6*eta(q^12)^6) in powers of q.
G.f.: theta_3(q^3)^3/theta_3(q).
G.f.: 1+2( Sum_{k>0} x^(3k-1)/(1-(-x)^(3k-1)) - x^(3k-2)/(1-(-x)^(3k-2))) = 1 +2( Sum_{k>0} (-1)^k x^k/(1+x^k+x^(2k)) +2 x^(4k)/(1+x^(4k)+x^(8k)) ).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Nov 14 2023

A123331 Expansion of (c(q)^2/(3c(q^2))-1)/2 in powers of q where c(q) is a cubic AGM function.

Original entry on oeis.org

1, 2, 1, 1, 0, 2, 2, 2, 1, 0, 0, 1, 2, 4, 0, 1, 0, 2, 2, 0, 2, 0, 0, 2, 1, 4, 1, 2, 0, 0, 2, 2, 0, 0, 0, 1, 2, 4, 2, 0, 0, 4, 2, 0, 0, 0, 0, 1, 3, 2, 0, 2, 0, 2, 0, 4, 2, 0, 0, 0, 2, 4, 2, 1, 0, 0, 2, 0, 0, 0, 0, 2, 2, 4, 1, 2, 0, 4, 2, 0, 1, 0, 0, 2, 0, 4, 0, 0, 0, 0, 4, 0, 2, 0, 0, 2, 2, 6, 0, 1, 0, 0, 2, 4, 0
Offset: 1

Views

Author

Michael Somos, Sep 26 2006

Keywords

Links

Crossrefs

Cf. A123330(n)=2*a(n) if n>0. A113974(n)=-(-1)^n*a(n).
Cf. A248897.

Programs

  • Mathematica
    f[p_, e_] := If[Mod[p, 6] == 1, e+1, (1+(-1)^e)/2]; f[2, e_] := (3-(-1)^e)/2; f[3, e_] := 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)
  • PARI
    {a(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*kronecker(-3,d)))}
  • PARI
    {a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==2, (3-(-1)^e)/2, if(p==3, 1, if(p%6==1, e+1, !(e%2)))))))}
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( (eta(x^2+A)*eta(x^3+A)^6/ eta(x+A)^2/eta(x^6+A)^3-1)/2, n))}

Formula

Moebius transform is period 6 sequence [ 1, 1, 0, -1, -1, 0, ...].
a(n) is multiplicative with a(2^e) = (3-(-1)^e)/2, a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 6), a(p^e) = (1+(-1)^e)/2 if p == 5 (mod 6).
a(3n) = a(4n) = a(n). a(6n+5) = 0.
G.f.: Sum_{k>0} x^k/(1-x^k+x^(2k)) = (theta_3(-q^3)^3/theta_3(-q) - 1)/2.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/(3*sqrt(3)) = 1.209199... (A248897). - Amiram Eldar, Nov 14 2023
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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