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.

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

A138522 Expansion of f(q, q^3)^2 / (f(q, q^4) * f(q^2, q^3)) in powers of q where f(, ) is the Ramanujan general theta function.

Original entry on oeis.org

1, 1, -1, 0, -1, 1, 4, -4, -1, -3, 3, 12, -12, -2, -8, 8, 31, -30, -5, -20, 19, 72, -68, -12, -44, 41, 154, -144, -24, -90, 84, 312, -289, -48, -178, 164, 603, -554, -92, -336, 307, 1122, -1024, -168, -612, 557, 2024, -1836, -300, -1087, 983, 3552, -3206, -522
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).

Examples

			G.f. = 1 + q - q^2 - q^4 + q^5 + 4*q^6 - 4*q^7 - q^8 - 3*q^9 + 3*q^10 + 12*q^11 + ...

Links

Crossrefs

Cf. A095813, A138520.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] / QPochhammer[ q^5])^3 QPochhammer[ q^10] / QPochhammer[ q], {q, 0, n}]; (* Michael Somos, Sep 14 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A) / eta(x + A) * ( eta(x^2 + A) / eta(x^5 + A) )^3, n))};

Formula

Expansion of (eta(q^2) / eta(q^5))^3 * eta(q^10) / eta(q) in powers of q.
Euler transform of period 10 sequence [ 1, -2, 1, -2, 4, -2, 1, -2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v)^2 - u * (u + 4) * (1 - v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = (u-v)^4 - u * (1 - u) * (4 + u) * v * (1 - v) * (4 + v).
G.f. is a period 1 Fourier series which satisfies f(-1 / (10 t)) = (5/4) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138520.
G.f.: Product_{k>0} P(2, x^k)^4 * P(10, x^k) / P(5, x^k)^2 where P(n, x) is the n-th cyclotomic polynomial.
A095813(n) = a(n) unless n=0. Convolution inverse of A138520.

A228864 Expansion of 1 + q * (psi(-q^5) / psi(-q))^2 in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 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, 321584, 391993, 476952
Offset: 0

Views

Author

Michael Somos, Sep 05 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 + 3*x^3 + 6*x^4 + 11*x^5 + 16*x^6 + 24*x^7 + 38*x^8 + ...

Links

Crossrefs

Cf. A138520, A210458, A225849.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 + (EllipticTheta[ 2, Pi/4, q^(5/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)])^2, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q^5] / EllipticTheta[ 3, 0, q])^2 QPochhammer[ q^5, -q^5] / QPochhammer[ q, -q]^5, {q, 0, n}]; (* Michael Somos, Oct 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^10 + A)^8 / (eta(x + A) * eta(x^4 + A) * eta(x^5 + A)^3 * eta(x^20 + A)^3), n))};

Formula

Expansion of (phi(q^5) / phi(q))^2 * (chi^5(q) / chi(q^5)) in powers of q where phi(), chi() are Ramanujan theta functions.
Expansion of eta(q^10)^8 / (eta(q) * eta(q^4) * eta(q^5)^3 * eta(q^20)^3) in powers of q.
Euler transform of period 20 sequence [ 1, 1, 1, 2, 4, 1, 1, 2, 1, -4, 1, 2, 1, 1, 4, 2, 1, 1, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = (1/5) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A225849.
a(n) = A210458(n) unless n=0. a(n) = (-1)^n * A138520(n).
a(n) ~ exp(2*Pi*sqrt(n/5)) / (2 * 5^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 15 2017
Showing 1-3 of 3 results.

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