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.

A143379 Expansion of q^(-7/24) * eta(q) * eta(q^4)^2 / eta(q^2) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, -1, 1, 1, 1, -1, 1, 0, 1, 0, 0, -2, -1, 0, 0, -1, 1, 1, -2, 0, 0, 0, 1, 1, 0, 2, 0, 1, -1, -1, 0, 1, -1, 0, 0, 1, 0, -1, -1, 0, -1, -1, -1, 0, 0, 0, 1, 0, 1, 0, 1, -1, -1, 2, 0, -1, 1, -1, 1, 0, 3, 1, -1, 0, 0, 0, 1, -2, 0, 0, -1, -1, 0, -1, 0, 1, 0, 0, 1, -1, -1, -1, 0, 0, 0, 0, -1, 0, -2, 0, 1, 2, 1, -1, 0, 2, 1, 0, 0, 0, 0, 1
Offset: 0

Views

Author

Michael Somos, Aug 11 2008

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^3 - x^4 + x^5 + x^6 + x^7 - x^8 + x^9 + x^11 - 2*x^14 - x^15 - x^18 + ...
G.f. = q^7 - q^31 - q^79 - q^103 + q^127 + q^151 + q^175 - q^199 + q^223 + ...

Links

Crossrefs

Cf. A000009, A134343, A143377, A143380.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^4]^2 / QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Jul 11 2012 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)
  • PARI
    {a(n) = my(A, p, e, x); if( n<0, 0, n = n*4 + 1; A = factor(6*n + 1); simplify( I^n / -2 * prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^2 / eta(x^2 + A), n))};

Formula

Expansion of psi(x^2) * f(-x) = psi(-x) * f(-x^4) = chi(-x) * f(-x^4)^2 = psi(-x)^2 / chi(-x) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015
Euler transform of period 4 sequence [ -1, 0, -1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 72^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143377.
G.f.: Product_{k>0} (1 - x^(4*k))^2 * (1 - x^(2*k-1)).
Convolution of A000009 and A134343. - Michael Somos, Jul 11 2012
-2 * a(n) = A143377(4*n + 1). 2 * a(n) = A143380(4*n + 1).
a(2*n) = A214302(n). a(2*n + 1) = - A214303(n). - Michael Somos, Jul 11 2012

A143378 Expansion of q^(-1/24) * eta(q^2)^5 / (eta(q) * eta(q^4)^2) in powers of q.

Original entry on oeis.org

1, 1, -3, -2, 2, -1, 0, 1, 2, 4, -2, 0, -1, -2, 2, -1, 0, -2, -2, -2, 0, 0, 1, 4, -2, 2, 1, 0, -2, 0, 4, 0, 2, 0, 0, 1, 0, -4, 0, -2, -3, 0, 2, 2, -4, 0, 0, 2, -2, 0, -2, -3, 2, 0, 2, 2, 0, 1, 4, 0, 0, 0, 2, 0, 0, -4, 0, 2, 0, 2, -1, 0, 0, 2, -2, 2, -2, -1, -2, -4, 0, 0, 0, -2, -2, 0, 0, 2, 2, -2, 2, 0, 1, 0, 0, -2, 0, 0, 0, -2, 5, 2, -4, 2, 0
Offset: 0

Views

Author

Michael Somos, Aug 11 2008

Keywords

Comments

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

Examples

			G.f. = 1 + x - 3*x^2 - 2*x^3 + 2*x^4 - x^5 + x^7 + 2*x^8 + 9*x^9 + ...
G.f. = q + q^25 - 3*q^49 - 2*q^73 + 2*q^97 - q^121 + q^169 + 2*q^193 + ...

Links

Crossrefs

Cf. A143377, A143380.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 3, 0, x], {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x] EllipticTheta[ 4, 0, x^2], {x, 0, n}]; (* Michael Somos, Apr 07 2015 *)
  • PARI
    {a(n) = my(A, p, e, x); if(n<0, 0, n *= 4; A = factor(6*n + 1); simplify( I^n * prod(k=1, matsize(A)[1], if(p = A[k, 1], e = A[k, 2]; if(p<5, 0, if(p%8==5 || p%24==23, !(e%2), if(p%8==3 || p%24==17, (-1)^(e\2) * !(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e)))))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A)^2), n))};

Formula

Expansion of phi(x) * f(-x) = phi(-x^2) * f(x) = chi(-x) * f(x)^2 = phi(-x^2)^2 / chi(-x) in powers of x where phi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 1, -4, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 288^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143380.
G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 - x^(2*k-1)).
a(n) = A143377(4*n) = A143380(4*n).

A143380 Expansion of q^(-1/6) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)) in powers of q.

Original entry on oeis.org

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

Views

Author

Michael Somos, Aug 11 2008

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + x^4 - 2*x^5 - 3*x^8 - 2*x^12 - 2*x^13 + 2*x^16 - 2*x^17 + ...
G.f. = q + 2*q^7 + q^25 - 2*q^31 - 3*q^49 - 2*q^73 - 2*q^79 + 2*q^97 + ...

Links

Crossrefs

Cf. A143377, A143378, A143379.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^5 / (QPochhammer[ x]^2 QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
  • PARI
    {a(n)= my(A, p, e, x); if(n<0, 0, A = factor(6*n + 1); simplify( I^-n * prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p<5, 0, p%8==5 || p%24==23, !(e%2), p%8==3 || p%24==17, (-1)^(e\2)*!(e%2), for(i=1, sqrtint(p\6), if( issquare(p - 6*i^2, &x), break)); (e+1) * (kronecker(12, x) * I^((p-1) / 6))^e ))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)), n))};

Formula

Expansion of psi(x)^2 * chi(-x^2) in powers of x where phi(), chi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ 2, -3, 2, -2, ...].
a(n) = (-1)^(-n / 2) * b(6*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = 0^e, b(p^e) = (1 + (-1)^e) / 2 if p == 5 (mod 8) or p == 23 (mod 24), b(p^e) = (-1)^(e/2) * (1 + (-1)^e) / 2 if p == 3 (mod 8) or p == 17 (mod 24) and p>3, b(p^e) = (e+1) * s^e if p == 1, 7 (mod 24) where p = x^2 + 6*y^2 and s = Kronecker(12, x) * (-1)^((p-1) / 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 1152^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143378.
a(4*n + 2) = a(4*n + 3) = 0.
G.f.: Product_{k>0} (1 - (-x)^k)^2 * (1 + x^(2*k)).
a(n) = (-1)^n * A143377(n). a(4*n) = A143378(n). a(4*n + 1) = 2 * A143379(n).
Showing 1-3 of 3 results.

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