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.

A083650 Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, May 01 2003

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Essentially the expansion of eta(q)*eta(q^2). Cf. A010815. - N. J. A. Sloane, Feb 18 2010
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

Examples

			G.f. = 1 - x + 2*x^2 - x^3 + 2*x^5 - x^6 - 2*x^9 - x^10 + 2*x^11 - 2*x^12 - 2*x^14 + ...
G.f. = q - q^9 + 2*q^17 - q^25 + 2*q^41 - q^49 + 2*q^73 - q^81 + 2*q^89 - 2*q^97 + ...

Links

Crossrefs

Cf. A000122, A000700, A010054, A010815, A030204, A083650, A121373, A138514, A143433.

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q]*QP[q^2] + O[q]^105; Table[(-1)^Quotient[n, 2]*Coefficient[s, q, n], {n, 0, 105}] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(x + A) * eta(x^2 + A), n))}; /* Michael Somos, Mar 02 2010 */

Formula

Euler transform of period 16 sequence [ -1, 2, 1, -2, 1, 1, -1, -3, -1, 1, 1, -2, 1, 2, -1, -2, ...].
G.f.: (Sum_{k>=0} (-1)^(k + [k/4]) * x^(k*(k+1)/2)) * (Sum_k x^(2*k^2)).
(-1)^[n/2] * a(n) = A030204(n).

Extensions

Revised by Michael Somos, Mar 02 2010

A138514 Expansion of q^(-1/8) * eta(q^2)^4 / (eta(q) * eta(q^4)) in powers of q.

Original entry on oeis.org

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

Views

Author

Michael Somos, Mar 22 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 70 of the 74 eta-quotients listed in Table I of Martin (1996).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

Examples

			G.f. = 1 + x - 2*x^2 - x^3 - 2*x^5 + x^6 + 2*x^9 + x^10 + 2*x^11 - 2*x^12 + ...
G.f. = q + q^9 - 2*q^17 - q^25 - 2*q^41 + q^49 + 2*q^73 + q^81 + 2*q^89 + ...

Links

Crossrefs

Cf. A030204, A083650.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2] QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^4 / (QPochhammer[ x] QPochhammer[ x^4]), {x, 0, n}]; (* Michael Somos, Jun 10 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e+1) * if( qfbclassno( -8 * p) / 4 % 2, (-1)^e, 1), if( e%2==0, (-1)^(e/2 * (p%8==5)))))) };
  • PARI
    {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([ 1, 0; 0, 64], n) - qfrep([ 4, 2; 2, 17], n))[n])};

Formula

Expansion of f(x) * f(-x^2) = psi(-x) * phi(x) = chi(x) * f(-x^2)^2 = psi(x) * phi(-x^2) = f(x)^2 / chi(x) = f(x)^3 / phi(x) = f(-x^2)^3 / psi(-x) = phi(x)^2 / chi(x)^3 = chi(x)^3 * psi(-x)^2 = (f(x)^3 * psi(-x))^(1/2) = (f(-x^2)^3 * phi(x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions.
Expansion of psi(i * x) * psi(-i * x) in powers x^2 where i^2 = -1 and psi() is a Ramanujan theta function. - Michael Somos, Feb 16 2014
Euler transform of period 4 sequence [ 1, -3, 1, -2, ...].
a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p == 3, 5, 7 (mod 8) and e odd, b(p^e) = 1 if p == 3 (mod 4) and e even, b(p^e) = (-1)^(e/2) if p == 5 (mod 8) and e even, b(p^e) = e+1 if p == 1 (mod 8) and p = x^2 + 64*y^2, b(p^e) = (-1)^e * (e+1) if p == 1 (mod 8) and p is not of the form x^2 + 64*y^2.
a(9*n + 1) = a(n), a(9*n + 4) = a(9*n + 7) = 0. a(n) = (-1)^n * A030204(n) = (-1)^floor((n+1)/2) * A083650(n).
G.f.: Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)).
G.f. is a period 1 Fourier series which satisfies f(-1 / (256 t)) = 16 (t/i) f(t) where q = exp(2 Pi i t). - Michael Somos, Jun 10 2015

A107063 Expansion of q^(-1/24) * (eta(q^2) * eta(q^3)^4) / (eta(q) * eta(q^6)^2) in powers of q.

Original entry on oeis.org

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

Views

Author

Michael Somos, May 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) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Links

Crossrefs

A030204(3*n) = a(n).

Programs

  • Mathematica
    eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-1/24)* (eta[q^2]*eta[q^3]^4)/(eta[q]*eta[q^6]^2), {q, 0, 100}], q] (* G. C. Greubel, Apr 18 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^4 / eta(x + A) / eta(x^6 + A)^2, n))}

Formula

Euler transform of period 6 sequence [1, 0, -3, 0, 1, -2, ...].
G.f.: Product_{k>0} (1+x^k)*(1-x^(3*k))^2/(1+x^(3*k))^2.
Expansion of phi(-q^3)^2 / chi(-q) in powers of q where phi(), chi() are Ramanujan theta functions.

A107064 Expansion of q^(-17/24) * (eta(q) * eta(q^6)^4) / (eta(q^2) * eta(q^3)^2) in powers of q.

Original entry on oeis.org

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

Views

Author

Michael Somos, May 10 2005

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).

Links

Crossrefs

Cf. A030204.

Programs

  • Mathematica
    eta[q_] := q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(-17/24)*(eta[q]*eta[q^6]^4)/(eta[q^2]*eta[q^3]^2), {q, 0, 100}], q] (* G. C. Greubel, Apr 18 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^4 / eta(x^2 + A) / eta(x^3 + A)^2, n))}

Formula

Euler transform of period 6 sequence [ -1, 0, 1, 0, -1, -2, ...].
G.f.: Product_{k>0} ((1-x^(6k))(1+x^(3k)))^2/(1+x^k).
Expansion of psi(q^3)^2 * chi(-q) in powers of q where psi(), chi() are Ramanujan theta functions.
2*a(n) = - A030204(3*n+2).

A230204 Expansion of phi(-x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 11 2013

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^3 + x^5 - 2*x^6 + 2*x^7 - 2*x^12 - x^14 - 2*x^15 + ...
G.f. = q - 2*q^17 + q^49 + q^81 - 2*q^97 + 2*q^113 - 2*q^193 - q^225 + ...

Links

Crossrefs

Cf. A030204.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] QPochhammer[ -q^3, q^8] QPochhammer[ -q^5, q^8] QPochhammer[ q^8], {q, 0, n}];
  • PARI
    {a(n) = local(m, j); if( n<0, 0, m = 16*n + 1; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==1 || (j%8==7)))))}

Formula

Euler transform of period 16 sequence [ -2, -1, -1, -1, -1, -2, -2, -2, -2, -2, -1, -1, -1, -1, -2, -2, ...].
a(n) = A030204(2*n).

A230205 Expansion of phi(-x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 11 2013

Keywords

Comments

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

Examples

			G.f. = 1 - x - 2*x^2 + 2*x^4 + 2*x^5 + x^7 - 2*x^8 - 2*x^9 - x^10 + ...
G.f. = q^9 - q^25 - 2*q^41 + 2*q^73 + 2*q^89 + q^121 - 2*q^137 - 2*q^153 + ...

Links

Crossrefs

Cf. A030204.

Programs

  • Mathematica
    a[ n_]:= SeriesCoefficient[EllipticTheta[4,0,q]*QPochhammer[-q^1,q^8]* QPochhammer[-q^7,q^8]*QPochhammer[q^8], {q, 0, n}];
  • PARI
    {a(n) = local(m, j); if( n<0, 0, m = 16*n + 9; sum( k=0, sqrtint(m \ 4), if( issquare(m - 16*k^2, &j), if( k==0, 1, 2) * (-1)^k * ((j%8)==3 || (j%8==5)))))}

Formula

Euler transform of period 16 sequence [ -1, -2, -2, -1, -2, -1, -1, -2, -1, -1, -2, -1, -2, -2, -1, -2, ...].
a(n) = - A030204(2*n + 1).

A319456 a(n) = [x^n] Product_{k>=1} ((1 - x^k)*(1 - x^(2*k)))^n.

Original entry on oeis.org

1, -1, -3, 14, -11, -81, 282, -57, -2043, 5405, 2417, -46476, 94522, 110512, -943407, 1505289, 2807589, -16888311, 23645199, 46006542, -265972791, 472882620, 187884672, -3981273597, 14234579226, -19187383356, -78662039004, 502118911904, -847583768679, -2627514175002
Offset: 0

Views

Author

Ilya Gutkovskiy, Sep 19 2018

Keywords

Crossrefs

Cf. A002171, A002288, A008705, A030204, A030211, A066535, A296043, A296044, A319457.

Programs

  • Mathematica
    Table[SeriesCoefficient[Product[((1 - x^k) (1 - x^(2 k)))^n , {k, 1, n}], {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[(QPochhammer[x] QPochhammer[x^2])^n, {x, 0, n}], {n, 0, 29}]
Table[SeriesCoefficient[Exp[n Sum[(DivisorSigma[1, 2 k] - 4 DivisorSigma[1, k]) x^k/k, {k, 1, n}]], {x, 0, n}], {n, 0, 29}]

Formula

a(n) = [x^n] Product_{k>=1} (1 - x^(2*k))^(2*n)/(1 + x^k)^n.
a(n) = [x^n] exp(n*Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k).
Showing 1-7 of 7 results.

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