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 1-8 of 8 results.

A098098 a(n) = sigma(6*n+5)/6.

Original entry on oeis.org

1, 2, 3, 4, 5, 8, 7, 8, 9, 10, 14, 12, 16, 14, 15, 20, 17, 18, 19, 24, 26, 22, 23, 28, 25, 32, 32, 28, 29, 30, 38, 32, 33, 40, 40, 44, 42, 38, 39, 40, 57, 42, 43, 44, 45, 62, 47, 56, 49, 56, 62, 52, 53, 60, 64, 68, 64, 58, 59, 60, 74, 72, 70, 64, 65, 80, 67, 76, 80, 70, 93, 72
Offset: 0

Views

Author

Vladeta Jovovic, Sep 14 2004

Keywords

Comments

Euler transform of period 6 sequence [2, 0, 0, 0, 2, -4, ...].
Expansion of q^(-5/6) * (eta(q)^-1 * eta(q^2) * eta(q^3) * eta(q^6))^2 in powers of q. - Michael Somos, Sep 16 2004
2*a(n) is the number of bipartitions of 2*n+1 that are 3-cores. See Baruah and Nath. - Michel Marcus, Apr 13 2020

Examples

			G.f. =1 + 2*x + 3*x^2 + 4*x^3 + 5*x^4 + 8*x^5 + 7*x^6 + 8*x^7 + 9*x^8 + 10*x^9 + ...
G.f. = q^5 + 2*q^11 + 3*q^17 + 4*q^23 + 5*q^29 + 8*q^35 + 7*q^41 + 8*q^47 + 9*q^53 + ...
		

Crossrefs

Programs

  • Magma
    Basis( ModularForms( Gamma0( 36), 2), 432)[6]; /* Michael Somos, Jul 09 2018 */
  • Mathematica
    Table[DivisorSigma[1, 6 n + 5]/6, {n, 0, 71}] (* Ivan Neretin, Apr 30 2016 *)
  • PARI
    a(n) = sigma(6*n + 5)/6
    
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A))^2, n))} /* Michael Somos, Sep 16 2004 */
    

Formula

G.f.: (Product_{k>0} (1 + x^k) * (1 - x^(3*k)) * (1 - x^(6*k)))^2. - Michael Somos, Sep 16 2004
From Michael Somos, Jul 09 2018: (Start)
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A252650. -
Convolution square of A121444.
A232343(2*n) = (-1)^n * A258831(n) = A000203(6*n + 4) = a(n). A033686(2*n) = -A134079(2*n + 1) = 2 * a(n). A121443(6*n + 5) = A133739(6*n + 5) = A232356(6*n + 5) = A134077(3*n + 2) = 6 * a(n). A125514(6*n + 5) = 24 * a(n). A134078(6*n + 5) = -36 * a(n). A186100(6*n + 5) = -72 * a(n). (End)
From Amiram Eldar, Dec 16 2022: (Start)
a(n) = A000203(A016969(n))/6.
Sum_{k=1..n} a(k) = (Pi^2/18) * n^2 + O(n*log(n)). (End)

A131943 Expansion of b(q) * b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, -3, 15, -3, -18, 15, -24, -3, 69, -18, -36, 15, -42, -24, 90, -3, -54, 69, -60, -18, 120, -36, -72, 15, -93, -42, 231, -24, -90, 90, -96, -3, 180, -54, -144, 69, -114, -60, 210, -18, -126, 120, -132, -36, 414, -72, -144, 15, -171, -93, 270, -42, -162
Offset: 0

Views

Author

Michael Somos, Jul 30 2007

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 3*q - 3*q^2 + 15*q^3 - 3*q^4 - 18*q^5 + 15*q^6 - 24*q^7 - 3*q^8 +...
		

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 84, Eq. (32.65).

Crossrefs

Programs

  • Magma
    A := Basis( ModularForms( Gamma0(6), 2), 54); A[1] - 3*A[2] - 3*A[3]; /* Michael Somos, Aug 30 2014 */
    
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q]^3 QPochhammer[ q^2]^3 / (QPochhammer[ q^3] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Nov 21 2013 *)
    a[ n_] := If[ n < 1, Boole[n == 0], -3 Sum[ d {0, 1, 0, -2, 0, 1}[[ Mod[ d, 6] + 1]], {d, Divisors @ n}]]; (* Michael Somos, Nov 11 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -3 * sumdiv(n, d, d * ((d%6==1) + (d%6==5) - 2 * (d%6==3))))};
    
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^2 + A))^3 / (eta(x^3 + A) * eta(x^6 + A)), n))};
    
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 1, p==3, 4 - 3^(e+1), (p^(e+1) - 1) / (p - 1) )))}; /* Michael Somos, Nov 21 2013 */
    
  • Sage
    A = ModularForms( Gamma0(6), 2, prec=54) . basis();  A[0] - 3*A[1] - 3*A[2]; # Michael Somos, Nov 21 2013

Formula

Expansion of eta(q)^3 * eta(q^2)^3 / (eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -6, -2, -6, -3, -4, ...].
a(n) = -3 * b(n) where b() is multiplicative with b(2^e) = 1, b(3^e) = 4 - 3^(e+1), b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3. - Michael Somos, Nov 21 2013
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = 54 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121443.
G.f.: Product_{k>0} ((1 - x^k) * (1 - x^(2*k)))^3 / ((1 - x^(3*k)) * (1 - x^(6*k))).
G.f.: 1 - 3 * (Sum_{k>0} (6*k - 1) * x^(6*k - 1) / (1 - x^(6*k - 1)) - 2*(6*k - 5) * x^(6*k - 3) / (1 - x^(6*k - 3)) + (6*k - 5) * x^(6*k - 5) / (1 -x^(6*k - 5))).
a(n) = a(2*n). a(n) = -3 * A131944(n) unless n=0. a(3^n) = 3 * A168611(n+1). a(2*n + 1) = -3 * A134077(n). - Michael Somos, Nov 21 2013

A186100 Expansion of 2 * a(q^2)^2 - a(q)^2 in powers of q where a() is a cubic AGM theta function.

Original entry on oeis.org

1, -12, -12, -12, -12, -72, -12, -96, -12, -12, -72, -144, -12, -168, -96, -72, -12, -216, -12, -240, -72, -96, -144, -288, -12, -372, -168, -12, -96, -360, -72, -384, -12, -144, -216, -576, -12, -456, -240, -168, -72, -504, -96, -528, -144, -72, -288
Offset: 0

Views

Author

Michael Somos, Feb 12 2011

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Ramanujan's Eisenstein series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

Examples

			G.f. = 1 - 12*q - 12*q^2 - 12*q^3 - 12*q^4 - 72*q^5 - 12*q^6 - 96*q^7 + ...
		

Crossrefs

Programs

  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], -12 DivisorSum[ n, # Boole[ 1 == GCD[#, 6]] &]]; (* Michael Somos, Jul 07 2015 *)
    a[ n_] := SeriesCoefficient[(EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^3])^2 - 1/2 (EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, 0, x^(3/2)])^2, {x, 0, n}]; (* Michael Somos, Jul 07 2015 *)
  • PARI
    {a(n) = if( n<1, n==0, -12 * sumdiv( n, d, d * (1 == gcd( d, 6) ) ) )};
    
  • PARI
    {a(n) = if( n<1, n==0, -12 * direuler( p=2, n, 1 / (1 - X) / (1 - (p>3) * p * X)) [n])};

Formula

Expansion of b(x) * b(x^2) - c(x) * c(x^2) in powers of x where b(), c() are cubic AGM functions.
Expansion of (phi(-x) * phi(-x^3))^2 - 8 * x * (psi(x) * psi(x^3))^2 in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of (P(q) - 2*P(q^2) - 3*P(q^3) + 6*P(q^6)) / 2 in powers of q where P() is a Ramanujan Eisenstein series. - Michael Somos, Jul 07 2015
a(n) = -12 * A186099(n) if n>0. a(2*n) = a(n). a(2*n + 1) = - A008653(2*n + 1). a(n) = 2 * A008653(n) - A008653(2*n) = A131946(n) - 8 * A111932(n) = A131943(n) - 9 * A121443(n).
a(3*n) = a(n). a(6*n + 5) = -72 * A098098(n).- Michael Somos, Jul 07 2015

A124449 Expansion of (phi(-q^3)^4 - phi(-q)^4)/8 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 3, -3, 6, -9, 8, -3, 9, -18, 12, -9, 14, -24, 18, -3, 18, -27, 20, -18, 24, -36, 24, -9, 31, -42, 27, -24, 30, -54, 32, -3, 36, -54, 48, -27, 38, -60, 42, -18, 42, -72, 44, -36, 54, -72, 48, -9, 57, -93, 54, -42, 54, -81, 72, -24, 60, -90, 60, -54, 62, -96, 72, -3, 84, -108, 68, -54, 72, -144, 72, -27, 74, -114
Offset: 1

Views

Author

Michael Somos, Nov 01 2006

Keywords

Comments

Ramanujan theta functions: f(q) := Product_{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) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Crossrefs

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[(EllipticTheta[3, 0, -q^3]^4 - EllipticTheta[ 3, 0, -q]^4)/8, {q, 0, n}]; Table[a[n], {n, 1, 50}] (* G. C. Greubel, Dec 16 2017 *)
  • 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, if(p==3, 3^e, (p^(e+1)-1)/(p-1))))))}
    
  • PARI
    {a(n)=local(A); if(n<1, 0, n--; A=x*O(x^n); polcoeff( eta(x+A)^3*eta(x^6+A)^5/ eta(x^2+A)^3/eta(x^3+A), n))}

Formula

Expansion of eta(q)^3*eta(q^6)^5/(eta(q^2)^3*eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, -4, ...].
a(n) is multiplicative with a(2^e) = -3 if e>0, a(3^e) = 3^e, a(p^e) =(p^(e+1)-1)/(p-1) if p>3.
G.f.: x*Product_{k>0} (1+x^(3k))*(1-x^(6k))^4/(1+x^k)^3.
a(3*n) = 3*a(n). a(4*n) = a(2*n).
A121443(n) = -a(2*n)/3.
Dirichlet g.f.: (1 - 1/2^(s-2)) * (1 - 1/2^(s-1)) * (1 - 1/3^s) * zeta(s-1) * zeta(s). - Amiram Eldar, Sep 12 2023

A185717 Expansion of q^(-1) * c(q^2) * (c(q) - c(q^4)) / 9 in powers of q^2 where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 6, 8, 9, 12, 14, 18, 18, 20, 24, 24, 31, 27, 30, 32, 36, 48, 38, 42, 42, 44, 54, 48, 57, 54, 54, 72, 60, 60, 62, 72, 84, 68, 72, 72, 74, 93, 96, 80, 81, 84, 108, 90, 90, 112, 96, 120, 98, 108, 102, 104, 144, 108, 110, 114, 114, 144, 126, 144, 133, 126, 156, 128
Offset: 0

Views

Author

Michael Somos, Feb 10 2011

Keywords

Comments

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

Examples

			1 + 3*x + 6*x^2 + 8*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 18*x^7 + 18*x^8 + ...
q + 3*q^3 + 6*q^5 + 8*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 18*q^15 + ...
		

Crossrefs

Programs

  • Mathematica
    A185717[n_] := SeriesCoefficient[(QPochhammer[q^3, q^3]/QPochhammer[-q^3, q^3])^4*(1/(QPochhammer[q, q^2]*QPochhammer[q^3, q^6])^3), {q, 0, n}];
    Table[A185717[n], {n, 0, 50}] (* G. C. Greubel, Jul 10 2017 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, if( (n/d) % 3, 1, 0) * d))}
    
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^5 / (eta(x + A)^3 * eta(x^6 + A)), n))}

Formula

Expansion of phi(-x^3)^4 / (chi(-x) * chi(-x^3))^3 in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 6 sequence [ 3, 0, -2, 0, 3, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 3^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 2 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A118271.
a(3*n + 1) = 3 * a(n). A078708(2*n + 1) = A121443(2*n + 1) = A124449(2*n + 1).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/9 = 1.0966227... (A100044). - Amiram Eldar, Dec 28 2023

A214361 Expansion of c(q^2) * (c(q) + 2 * c(q^4)) / 9 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 3, 3, 6, 9, 8, 3, 9, 18, 12, 9, 14, 24, 18, 3, 18, 27, 20, 18, 24, 36, 24, 9, 31, 42, 27, 24, 30, 54, 32, 3, 36, 54, 48, 27, 38, 60, 42, 18, 42, 72, 44, 36, 54, 72, 48, 9, 57, 93, 54, 42, 54, 81, 72, 24, 60, 90, 60, 54, 62, 96, 72, 3, 84, 108, 68, 54
Offset: 1

Views

Author

Michael Somos, Jul 18 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
b(n) = 6*a(n) is the number of solutions in integers to n = x^2 + y^2 + z^2 + w^2 where x + y + z is not divisible by 3. - Michael Somos, Jun 23 2018

Examples

			G.f. = q + 3*q^2 + 3*q^3 + 3*q^4 + 6*q^5 + 9*q^6 + 8*q^7 + 3*q^8 + 9*q^9 + 18*q^10 + ...
a(1) = 1, b(1) = 6 with solutions (w, x, y, z) = {(0, 0, 1, 0), (0, 1, 0, 0), (1, 0, 0, 0)} and their negatives. - _Michael Somos_, Jun 23 2018
		

Crossrefs

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^4 - EllipticTheta[ 3, 0, q^3]^4) / 8, {q, 0, n}];
    a[ n_] := SeriesCoefficient[ (QPochhammer[ -q, q^2] QPochhammer[ -q^3, q^6])^3 EllipticTheta[ 2, 0, (-q)^(3/2)]^4 / (-16 (-q)^(1/2)), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^6 + A)^2 * eta(x^12 + A) / ( eta(x + A) * eta(x^4 + A))^3, n))};
    
  • PARI
    {a(n) = if( n<1, 0, sigma(n) + if( n%3==0, -1 * sigma(n/3)) + if( n%4==0, -4 * sigma(n/4)) + if( n%12==0, +4 * sigma(n/12)))};
    
  • PARI
    {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 3, p==3, 3^e, (p^(e+1) - 1) / (p - 1))))};

Formula

Expansion of (phi(q)^4 - phi(q^3)^4) / 8 = q * phi(q^3) * (chi(q) * psi(-q^3))^3 = q * psi(-q^3)^4 * (chi(q) * chi(q^3))^3 in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q^2)^6 * eta(q^3) * eta(q^6)^2 * eta(q^12) / ( eta(q) * eta(q^4))^3 in powers of q.
Euler transform of period 12 sequence [3, -3, 2, 0, 3, -6, 3, 0, 2, -3, 3, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A214456.
a(n) is multiplicative with a(2^e) = 3 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f.: x * Product_{k>0} (1 + (-x)^(3*k)) * (1 - x^(6*k))^4 / ( 1 + (-x)^k)^3.
a(n) = -(-1)^n * A124449(n). a(3*n) = 3*a(n). a(2*n) = a(4*n) = 3 * A121443(n). a(2*n + 1) = A185717(n).
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(2*s-2)) * (1 - 1/3^s) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/18 = 0.548311... (A086463). (End)

A226132 Expansion of - c(-q) * c(q^2) / 9 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, -1, 3, -1, 6, -3, 8, -1, 9, -6, 12, -3, 14, -8, 18, -1, 18, -9, 20, -6, 24, -12, 24, -3, 31, -14, 27, -8, 30, -18, 32, -1, 36, -18, 48, -9, 38, -20, 42, -6, 42, -24, 44, -12, 54, -24, 48, -3, 57, -31, 54, -14, 54, -27, 72, -8, 60, -30, 60, -18, 62, -32, 72
Offset: 1

Views

Author

Michael Somos, May 27 2013

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 91 of the 126 eta-quotients listed in Table 1 of Williams 2012.

Examples

			G.f. = q - q^2 + 3*q^3 - q^4 + 6*q^5 - 3*q^6 + 8*q^7 - q^8 + 9*q^9 - 6*q^10 + ...
		

Crossrefs

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ If[ OddQ[d] && ! Divisible[ n/d, 3], -d (-1)^(n/d), 0], {d, Divisors[ n]}]];
    a[ n_] := If[ n < 2, Boole[n == 1], Times @@ (Which[ # == 2, -1, # == 3, #^#2, True, (#^(#2 + 1) - 1) / (# - 1)] & @@@ FactorInteger[n])];
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3]^3 / EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, I q^(3/2)]^3 / EllipticTheta[ 2, 0, I q^(1/2)] / (4 (-1)^(1/4)), {q, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, if(d%2 && (n/d)%3, -d * (-1)^(n/d))))};
    
  • PARI
    {a(n) = my(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, -1, if( p==3, p^e, (p^(e+1) - 1) / (p - 1))))))};
    
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^12 / (eta(x^2 + A)^4 * eta(x^3 + A)^3 * eta(x^12 + A)^3), n))};

Formula

Expansion of (a(q) - a(q^2)) * (a(q^2) + 2 * a(q^4)) / 18 = c(q^2)^4 / (9 * c(q) * c(q^4)) = (b(-q) * b(q^2) - a(-q) * a(q^2)) / 9 in powers of q where a(), b(), c(q) are cubic AGM theta functions.
Expansion of q * (phi(q^3)^3 / phi(q)) * (ps(-q^3)^3 / psi(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^4) * eta(q^6)^12 / (eta(q^2)^4 * eta(q^3)^3 * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [ -1, 3, 2, 2, -1, -6, -1, 2, 2, 3, -1, -4, ...].
Multiplicative with a(2^e) = -1 if e>0, a(3^e) = 3^e, a(p^e) = (p^(e+1) - 1) / (p - 1) if p>3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4/3 (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A226139.
G.f.: Sum_{k>0 not 3|k} x^k / (1 - (-x)^k)^2 = Sum_{k>0 not 2|k} k * x^k * (1 - x^k) / (1 + x^(3*k)).
G.f.: x * Product_{k>0} (1 - x^k) * (1 - x^(4*k)) * (1 - x^(3*k))^6 * (1 + x^(3*k))^9 / ((1 - x^(2*k))^4 * (1 + x^(6*k))^3).
a(2*n) = - A121443(n). a(2*n + 1) = A185717(n).
a(n) = -(-1)^n * A121443(n). Convolution of A113447 and A113973.
From Amiram Eldar, Sep 12 2023: (Start)
Dirichlet g.f.: (1 - 1/2^(s-1))^2 * (1 - 1/3^s) * zeta(s-1) * zeta(s).
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^2/54 = 0.18277045... . (End)

A232356 Expansion of 2/9 * c(q) * c(q^2) - q * (psi(q) * psi(q^3))^2 in powers of q where psi() is a Ramanujan theta function and c(q) is a cubic AGM theta function.

Original entry on oeis.org

1, 0, 5, -2, 6, 4, 8, -6, 17, 0, 12, 2, 14, 0, 30, -14, 18, 16, 20, -12, 40, 0, 24, -2, 31, 0, 53, -16, 30, 24, 32, -30, 60, 0, 48, 14, 38, 0, 70, -36, 42, 32, 44, -24, 102, 0, 48, -10, 57, 0, 90, -28, 54, 52, 72, -48, 100, 0, 60, 12, 62, 0, 136, -62, 84, 48
Offset: 1

Views

Author

Michael Somos, Nov 22 2013

Keywords

Comments

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. = q + 5*q^3 - 2*q^4 + 6*q^5 + 4*q^6 + 8*q^7 - 6*q^8 + 17*q^9 + ...
		

Crossrefs

Programs

  • Magma
    Basis( ModularForms( Gamma0(6), 2), 70) [2];
  • Mathematica
    a[ n_] := If[ n < 1, 0, Sum[ d ( 2 Mod[ d, 2] Boole[Mod[ n/d, 3] > 0] - Mod[ n/d, 2] Boole[ Mod[d, 3] > 0]), {d, Divisors @n}]];
    a[ n_] := SeriesCoefficient[ 2 q (QPochhammer[ q^3] QPochhammer[ q^6])^3 / (QPochhammer[ q] QPochhammer[ q^2]) - q (QPochhammer[ q^2] QPochhammer[ q^6])^4 / (QPochhammer[ q] QPochhammer[ q^3])^2, {q, 0, n}];
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A=x*O(x^n); polcoeff( 2 * (eta(x^3 + A) * eta(x^6 + A))^3 / (eta(x + A) * eta(x^2 + A)) - (eta(x^2 + A) * eta(x^6 + A))^4 / (eta(x + A) * eta(x^3 + A))^2, n))};
    
  • Sage
    ModularForms( Gamma0(6), 2, prec=70).1;
    

Formula

a(n) = 2 * A121443(n) - A111932(n). a(2*n) = -2 * A229615(n). a(12*n + 2) = a(12*n + 10) = 0.
a(n) = A123532(n) + 7 * A229615(n). a(3*n + 2) = 6 * A232343(n-1). a(6*n + 5) = 6 * A098098(n). a(12*n + 4) = -2 * A144614(n). a(12*n + 6) = 4 * A008438(n). a(12*n + 8) = -6 * A033686(n). - Michael Somos, May 23 2014
Showing 1-8 of 8 results.