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 1-5 of 5 results.

A285348 Expansion of r(q^2) / r(q)^2 in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, 2, 0, -4, -2, 6, 8, -4, -16, -6, 20, 24, -12, -44, -16, 52, 62, -28, -108, -40, 122, 144, -64, -244, -88, 266, 308, -136, -508, -180, 544, 624, -272, -1008, -356, 1060, 1206, -524, -1920, -672, 1988, 2244, -968, -3524, -1224, 3606, 4048, -1732, -6284
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2017

Keywords

Comments

Let k(q) = r(q) * r(q^2)^2.
G.f. satisfies: A(q) = (1 + k(q))/(1 - k(q)).
And r(q^2)^5 = k(q)^2 * A(q).

Links

Crossrefs

r(q^k) / r(q)^k: this sequence (k=2), A285583 (k=3), A285584 (k=4), A285585 (k=5).
Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A112803 (1 + k(q)), A285349, A285355 (k(q)^2).

Formula

a(n) = A285349(n) - A138518(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

A285349 Expansion of r(q)^2 / r(q^2) in powers of q where r() is the Rogers-Ramanujan continued fraction.

Original entry on oeis.org

1, -2, 4, -4, 2, 2, -8, 12, -12, 6, 8, -24, 36, -36, 16, 20, -62, 92, -88, 40, 46, -144, 208, -196, 88, 102, -308, 440, -412, 180, 208, -624, 884, -816, 356, 404, -1206, 1692, -1552, 672, 760, -2244, 3128, -2852, 1224, 1378, -4048, 5612, -5084, 2174, 2428, -7104, 9796, -8836, 3760
Offset: 0

Views

Author

Seiichi Manyama, Apr 17 2017

Keywords

Comments

Let k(q) = r(q) * r(q^2)^2.
G.f. satisfies: A(q) = (1 - k(q))/(1 + k(q)).
And r(q)^5 = k(q) * A(q)^2.

Links

Crossrefs

r(q)^k / r(q^k): this sequence (k=2), A285628 (k=3), A285629 (k=4), A285630 (k=5).
Cf. A007325, A078905 (r(q)^5), A112274 (k(q)), A285348.

Formula

a(n) = A138518(n) + A285348(n) for n>0 (conjectured). - Thomas Baruchel, May 14 2018

A138519 Expansion of q * (psi(q^5) / psi(q))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 3, -6, 11, -16, 24, -38, 57, -82, 117, -168, 238, -328, 448, -614, 834, -1114, 1480, -1966, 2592, -3384, 4398, -5704, 7361, -9436, 12045, -15344, 19470, -24576, 30922, -38822, 48576, -60548, 75259, -93342, 115454, -142360, 175104, -214958, 263262
Offset: 1

Views

Author

Michael Somos, Mar 23 2008

Keywords

Comments

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

Examples

			G.f. = q - 2*q^2 + 3*q^3 - 6*q^4 + 11*q^5 - 16*q^6 + 24*q^7 - 38*q^8 + 57*q^9 + ...

Links

Crossrefs

Cf. A138516, A138520, A138521, A210458.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 2, 0, q^(5/2)] / EllipticTheta[ 2, 0, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Sep 16 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( ( eta(x + A) / eta(x^5 + A) * ( eta(x^10 + A) / eta(x^2 + A) )^2)^2, n))};

Formula

Expansion of ((eta(q^10) / eta(q^2))^2 * eta(q) / eta(q^5))^2 in powers of q.
Euler transform of period 10 sequence [ -2, 2, -2, 2, 0, 2, -2, 2, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - v * (1 - u) * (1 - 5*u).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138518.
G.f.: x * (Product_{k>0} P(5, x^k) * P(10, x^k)^2)^2 where P(n, x) is the n-th cyclotomic polynomial.
a(n) = - A138520(n) unless n=0. -5 * a(n) = A138521(n) unless n=0.
Convolution inverse of A138516.
a(n) = -(-1)^n * A210458(n). - Michael Somos, Sep 16 2015
a(n) ~ -(-1)^n * exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017

A138527 Expansion of phi(-q) / phi(-q^5) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 0, 0, 2, 2, -4, 0, 0, 2, 4, -8, 0, 0, 4, 8, -14, 0, 0, 8, 14, -24, 0, 0, 12, 22, -40, 0, 0, 20, 36, -64, 0, 0, 32, 56, -98, 0, 0, 48, 84, -148, 0, 0, 72, 126, -220, 0, 0, 106, 184, -320, 0, 0, 152, 264, -460, 0, 0, 216, 376, -652, 0, 0, 306, 528, -912, 0, 0, 424, 732, -1264, 0, 0, 584, 1008
Offset: 0

Views

Author

Michael Somos, Mar 23 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Denoted by t in Andrews and Berndt 2005. - Michael Somos, Apr 25 2016

Examples

			G.f. = 1 - 2*q + 2*q^4 + 2*q^5 - 4*q^6 + 2*q^9 + 4*q^10 - 8*q^11 + 4*q^14 + ...

References

  • G. E. Andrews and B. C. Berndt, Ramanujan's lost notebook, Part I, Springer, New York, 2005, MR2135178 (2005m:11001) See p. 337.

Links

Crossrefs

Cf. A116494, A138158, A138526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] / EllipticTheta[ 4, 0, q^5], {q, 0, n}]; (* Michael Somos, Sep 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) / eta(x^5 + A))^2 * eta(x^10 + A) / eta(x^2 + A), n))};

Formula

Expansion of (eta(q) / eta(q^5))^2 * eta(q^10) / eta(q^2) in powers of q.
Euler transform of period 10 sequence [ -2, -1, -2, -1, 0, -1, -2, -1, -2, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - u^2)^2 - u^2 * (1 - v^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (v^2 - u^2) * (u + v)^2 - u * v * (1 - u^2) * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u + v)^2 * w^2 - u * v * (5 - v^2).
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u2 * u3 - u1 * u6)^2 - u1 * u3 * (u6^2 - u2^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116494.
G.f.: Product_{k>0} P(10, x^k) / P(5, x^k) where P(n, x) is the n-th cyclotomic polynomial.
a(5*n + 2) = a(5*n + 3) = 0.
Convolution inverse is A138526. Convolution square is A138518.

A138517 Expansion of (phi(-q^5) / phi(-q))^2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 4, 12, 32, 76, 164, 336, 656, 1228, 2228, 3932, 6768, 11408, 18872, 30688, 49152, 77644, 121096, 186684, 284720, 429916, 643168, 953904, 1403312, 2048784, 2969764, 4275656, 6116480, 8696864, 12294680, 17285776, 24176288, 33645132
Offset: 0

Views

Author

Michael Somos, Mar 23 2008

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).

Examples

			1 + 4*q + 12*q^2 + 32*q^3 + 76*q^4 + 164*q^5 + 336*q^6 + 656*q^7 + ...

Links

Crossrefs

Cf. 4 * A095846(n) = a(n) unless n=0. Convolution inverse of A138518. Convolution square of A138526.

Programs

  • Mathematica
    eta[x_] := x^(1/24)*QPochhammer[x]; A138517[n_] := SeriesCoefficient[ ((eta[q^5]/eta[q])^2*eta[q^2]/eta[q^10])^2, {q, 0, n}]; Table[ A138517[n], {n, 0, 50}] (* G. C. Greubel, Sep 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( ((eta(x^5 + A) / eta(x + A))^2 * eta(x^2 + A) / eta(x^10 + A))^2, n))}

Formula

Expansion of ( (eta(q^5) / eta(q))^2 * eta(q^2) / eta(q^10) )^2 in powers of q.
Euler transform of period 10 sequence [ 4, 2, 4, 2, 0, 2, 4, 2, 4, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v^2) * (u - 1) - 4 * u * v * (v - 1).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u - v)^4 - u * (1 - u) * (1 - 5*u) * v * (1 - v) * (1 - 5*v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is g.f. for A138516.
G.f.: (Product_{k>0} P(5, x^k) / P(10, x^k))^2 where P(n, x) is the n-th cyclotomic polynomial.
a(n) ~ exp(2*Pi*sqrt(2*n/5)) / (2^(3/4) * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Jun 03 2018
Empirical: Sum_{n>=0} a(n)/exp(2*Pi*n) = 3/10 + (1/10)*sqrt(5) + (1/10)*sqrt(10 + 6*sqrt(5)). - Simon Plouffe, Mar 04 2021
Showing 1-5 of 5 results.

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