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.

A113419 Expansion of phi(x)^2 * phi(-x) * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, -4, -8, 7, 10, -12, -8, 18, 18, -16, -24, 21, 20, -28, -32, 20, 32, -36, -24, 42, 42, -28, -48, 57, 36, -52, -40, 36, 58, -60, -56, 48, 66, -48, -72, 74, 42, -80, -80, 61, 82, -72, -56, 90, 96, -64, -72, 98, 70, -100, -104, 64, 106, -108, -72, 114, 96
Offset: 0

Views

Author

Michael Somos, Oct 29 2005

Keywords

Comments

Number 46 of the 74 eta-quotients listed in Table I of Martin (1996).
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 + 7*x^4 + 10*x^5 - 12*x^6 - 8*x^7 + 18*x^8 + ...
G.f. = q + 2*q^3 - 4*q^5 - 8*q^7 + 7*q^9 + 10*q^11 - 12*q^13 - 8*q^15 + ...

Links

Crossrefs

Cf. A109039, A113417, A209940, A258096.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^9 QPochhammer[ q^8]^2 / (QPochhammer[ q]^2 QPochhammer[ q^4]^5), {q, 0, n}]; (* Michael Somos, May 19 2015 *)
a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv( n, d, d*(d%2) * (-1)^((n/d)\2 + (d-1)\4)))};
  • PARI
    {a(n) = my(A, p, e, t); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, t = (-1)^(p\2); p *= kronecker( -2, p); (p^(e+1) - t^(e+1)) / (p - t) )))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^8 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^5), n))};

Formula

Expansion of q^(-1/2) * (eta(q^2)^9 * eta(q^8)^2) / (eta(q)^2 * eta(q^4)^5) in powers of q. - Michael Somos, Mar 14 2012
Euler transform of period 8 sequence [ 2, -7, 2, -2, 2, -7, 2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (x^(e+1) - y^(e+1)) / (x - y) where x = p * (-1)^floor(p/4) and y = (-1)^floor(p/2).
G.f.: Sum_{k>0} (2*k - 1) * (-1)^[(k - 1)/2] * x^(2*k - 1) / (1 + x^(4*k - 2)).
a(n) = (-1)^n * A113417(n) = (-1)^floor(n/2) * A258096(n) = (-1)^(n + floor(n/2)) * A209940(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A109039. - Michael Somos, May 20 2015

A109040 Expansion of 1-eta(q)*eta(q^3)*(eta(q^4)*eta(q^6))^2/eta(q^12)^2 in powers of q.

Original entry on oeis.org

1, 1, 1, 1, -4, 1, -6, 1, 1, -4, 12, 1, 14, -6, -4, 1, -16, 1, -18, -4, -6, 12, 24, 1, 21, 14, 1, -6, -28, -4, -30, 1, 12, -16, 24, 1, 38, -18, 14, -4, -40, -6, -42, 12, -4, 24, 48, 1, 43, 21, -16, 14, -52, 1, -48, -6, -18, -28, 60, -4, 62, -30, -6, 1, -56, 12, -66, -16, 24, 24, 72, 1, 74, 38, 21, -18, -72, 14, -78, -4, 1
Offset: 1

Views

Author

Michael Somos, Jun 17 2005

Keywords

References

  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.68).

Crossrefs

Cf. A109039(n)=-a(n), if n>0.

Programs

  • Mathematica
    f[p_, e_] := If[MemberQ[{1, 11}, Mod[p, 12]], (p^(e+1)-1)/(p-1), ((-p)^(e+1)-1)/(-p-1)]; f[2, e_] := 1; f[3, e_] := 1; a[n_] := Times @@ f @@@ FactorInteger[n]; a[1] = 1; Array[a, 100] (* Amiram Eldar, Sep 05 2023 *)
  • PARI
    a(n)=if(n<1, 0, direuler(p=2,n,1/(1-X)/(1-p*kronecker(12,p)*X))[n])
  • PARI
    {a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( 1-eta(x+A)*eta(x^3+A)*eta(x^4+A)^2*eta(x^6+A)^2/eta(x^12+A)^2, n))}
  • PARI
    my(N=99, q='x+O('x^N)); Vec(1-eta(q)*eta(q^3)*(eta(q^4)*eta(q^6))^2/eta(q^12)^2) \\ Joerg Arndt, Sep 05 2023

Formula

Multiplicative with a(2^e)=a(3^e)=1, a(p^e)=(p^(e+1)-1)/(p-1) if p = 1, 11 (mod 12), a(p^e)=((-p)^(e+1)-1)/(-p-1) if p = 5, 7 (mod 12).
G.f.: 1-Product_{k>0} (1-x^k)(1-x^(3k))(1-x^(4k))^2/(1+x^(6k))^2 = Sum_{k>0} x^k*(1-3*x^(2k)+x^(4k))*(1+x^(2k))^3/(1+x^(6k))^2.
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024

A257656 Expansion of f(x) * f(x^3) * f(-x^4)^2 * chi(-x^6)^2 in powers of x where chi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -1, 1, -1, -4, -1, -6, -1, 1, 4, 12, -1, 14, 6, -4, -1, -16, -1, -18, 4, -6, -12, 24, -1, 21, -14, 1, 6, -28, 4, -30, -1, 12, 16, 24, -1, 38, 18, 14, 4, -40, 6, -42, -12, -4, -24, 48, -1, 43, -21, -16, -14, -52, -1, -48, 6, -18, 28, 60, 4, 62, 30, -6, -1
Offset: 0

Views

Author

Michael Somos, Jul 25 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x - x^2 + x^3 - x^4 - 4*x^5 - x^6 - 6*x^7 - x^8 + x^9 + 4*x^10 + ...

Links

Crossrefs

Cf. A109039.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x] QPochhammer[ -x^3] QPochhammer[ x^4]^2 QPochhammer[ x^6, x^12]^2, {x, 0, n}];
a[ n_] := If[ n<1, Boole[n == 0], Times @@ (If[ # < 5, -(-1)^#, With[ {t = # KroneckerSymbol[ 12, #]}, (t^(#2 + 1) - 1 ) / (t - 1)]]& @@@ FactorInteger @ n)];
  • PARI
    {a(n) = if( n<1, n==0, -sumdiv(n, d, d * kronecker( 12, d) * (-1)^(n/d)))};
  • 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^6 + A)^5 / (eta(x + A) * eta(x^3 + A) * eta(x^12 + A)^3), n))};
  • PARI
    {a(n) = my(A, p, e, t); if( n<1, n==0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, -(-1)^p, t = p * kronecker( 12, p); (t^(e+1) - 1) / (t - 1))))};

Formula

Expansion of eta(q^2)^3 * eta(q^4) * eta(q^6)^5 / (eta(q) * eta(q^3) * eta(q^12)^3) in powers of q.
Euler transform of period 12 sequence [1, -2, 2, -3, 1, -6, 1, -3, 2, -2, 1, -4, ...].
a(n) = a(3*n) = (-1)^n * A109039(n). a(2*n) = A109039(n).
Multiplicative with a(2^e) = -1, a(p^e) = ((p*Kronecker(12, p))^(e+1) - 1)/(p*Kronecker(12, p) - 1) for odd prime p. - Andrew Howroyd, Jul 27 2018
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024
Showing 1-3 of 3 results.

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