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 1-4 of 4 results.

A121373 Expansion of f(x) = f(x, -x^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 24 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = -x^3, b = -x. - Michael Somos, Jul 11 2012
Number 5 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 + x - x^2 - x^5 - x^7 - x^12 + x^15 + x^22 + x^26 + x^35 + ...
G.f. = q + q^25 - q^49 - q^121 - q^169 - q^289 + q^361 + q^529 + ...

Links

Crossrefs

Cf. A010815, A080995, A247133, A247223.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 - (-x)^k, {k, n}], {x, 0, n}]; (* Michael Somos, Nov 14 2011 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x], {x, 0, n}]; (* Michael Somos, Jul 06 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 1, Pi/12, x^4] + EllipticTheta[ 2, Pi/12, x^4]) / Sqrt[6], {x, 0, 24 n + 1}] // Simplify; (* Michael Somos, Mar 20 2015 *)
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), kronecker( 6, n))};
  • PARI
    {a(n) = if( n<0, 0, polcoeff( eta( -x + x * O(x^n)), n))};

Formula

Expansion of q^(-1/4) * (theta_1( Pi/12, q) + theta_2( Pi/12, q)) / sqrt(6) in powers of q^6. - Michael Somos, Jul 06 2013
Expansion of q^(-1/24) * eta(q^2)^3 / (eta(q) * eta(q^4)) in powers of q.
Euler transform of period 4 sequence [1, -2, 1, -1, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(p^2e) = (-1)^e if p == 7, 11, 13, 17 (mod 24), b(p^2e) = +1 if p == 1, 5, 19, 23 (mod 24) and b(p^(2e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f.: (1 + x) * (1 - x^2) * (1 + x^3) * (1 - x^4) * ...
G.f.: 1 + x - x^2*(1 + x) + x^3*(1 + x)*(1 - x^2) - x^4*(1 + x)*(1 - x^2)*(1 + x^3) + ...
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = a(n).
G.f.: Sum_{k>=0} a(k) * x^(24*k + 1) = Sum_{k in Z} (-1)^floor((k+1)/2) * x^(6*k + 1)^2.
a(n) = (-1)^n * A010815(n). |a(n)| = A080995(n).
Expansion of f(-x^5, -x^7) + x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 48^(1/2) (t/i)^(1/2) f(t) where q = exp(2 Pi i t). - Michael Somos, May 05 2016
G.f.: exp(Sum_{k>=1} (-1)^(k+1)*x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 08 2018

A133985 Expansion of f(-x, x^2) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 01 2007, Oct 04 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
a(n) is nonzero if and only if n is a number of A001318.
The exponents in the q-series for this sequence are the squares of the numbers of A007310.
Number 14 of the 14 primitive eta-products which are holomorphic modular forms of weight 1/2 listed by D. Zagier on page 30 of "The 1-2-3 of Modular Forms". - Michael Somos, May 04 2016

Examples

			G.f. = 1 - x + x^2 - x^5 - x^7 + x^12 - x^15 + x^22 + x^26 - x^35 + x^40 + ...
G.f. = q - q^25 + q^49 - q^121 - q^169 + q^289 - q^361 + q^529 + q^625 + ...

Links

Crossrefs

Cf. A001318, A007310, A080995, A133988, A247133, A247223.

Programs

  • Mathematica
    a[ n_] := (-1)^n Boole[ IntegerQ[ Sqrt[24 n + 1]]]; (* Michael Somos, Jan 10 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3]  QPochhammer[ x, -x], {x, 0, n}]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = (-1)^n * issquare( 24*n + 1) };
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^5 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A))^2, n))};

Formula

Expansion of phi(x^3) / chi(x) in powers of x where phi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/24) * eta(q) * eta(q^4) * eta(q^6)^5 / (eta(q^2) * eta(q^3) * eta(q^12))^2 in powers of q.
Euler transform of period 12 sequence [ -1, 1, 1, 0, -1, -2, -1, 0, 1, 1, -1, -1, ...].
a(n) = b(24*n + 1) where b() is multiplicative with b(p^(2*e)) = (-1)^e if p == 3, 5 (mod 8), b(p^(2*e)) = +1 if p == 1, 7 (mod 8) and b(p^(2*e-1)) = b(2^e) = b(3^e) = 0 if e>0.
G.f. is a period 1 Fourier series which satisfies f(-1 / (2304 t)) = 4 (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A133988.
a(5*n + 3) = a(5*n + 4) = 0. a(25*n + 1) = -a(n). a(n) = (-1)^n * A080995(n).
G.f. Sum_{k>=0} a(k) * q^(24*k + 1) = Sum_{k in Z} (-1)^floor(k/2) * q^(6*k + 1)^2.
Expansion of f(-x^5, -x^7) - x * f(-x, -x^11) in powers of x. - Michael Somos, Jan 10 2015

A186741 Expansion of f(x^5, x^7) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Jan 21 2012

Keywords

Comments

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

Examples

			G.f. = 1 + x^5 + x^7 + x^22 + x^26 + x^51 + x^57 + x^92 + x^100 + x^145 + ...
G.f. = q + q^121 + q^169 + q^529 + q^625 + q^1225 + q^1369 + q^2209 + q^2401 + ...

Links

Crossrefs

Cf. A010815, A036498, A247223.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[ QPochhammer[-q^5,q^12]*QPochhammer[-q^7,q^12] *QPochhammer[q^12,q^12], {q, 0, n}]; (* G. C. Greubel, Dec 08 2017 *)
  • PARI
    {a(n) = my(m); if( !issquare( 24*n + 1, &m), 0, m%12 == 1 || m%12 == 11)};

Formula

Euler transform of period 24 sequence [ 0, 0, 0, 0, 1, 0, 1, 0, 0, -1, 0, -1, 0, -1, 0, 0, 1, 0, 1, 0, 0, 0, 0, -1, ...].
a(n) is the characteristic function of A036498. a(n) = max( 0, A010815(n)).
G.f.: Sum_{k in Z} x^(6*k^2 - k) = Product_{k>0} (1 + x^(12*k - 7)) * (1 + x^(12*k - 5)) * (1 - x^(12*k)).
Sum_{k=1..n} a(k) ~ sqrt(2*n/3). - Amiram Eldar, Jan 13 2024

A308400 Expansion of 1 / Sum_{k=-oo..oo} (-x)^(k*(6*k + 1)).

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 1, 0, 2, 0, 1, 1, 0, 3, 0, 3, 1, 1, 3, 0, 6, 1, 3, 3, 1, 8, 1, 8, 3, 3, 9, 2, 14, 3, 9, 9, 4, 19, 4, 19, 9, 10, 21, 6, 32, 10, 22, 22, 12, 42, 12, 43, 23, 25, 48, 18, 67, 25, 51, 51, 31, 88, 31, 90, 54, 59, 101, 44, 137, 60, 108, 109, 73, 177, 73
Offset: 0

Views

Author

Ilya Gutkovskiy, May 24 2019

Keywords

Comments

Number of partitions of n into parts congruent to {0, 5, 7} mod 12.
Convolution inverse of A247223.

Crossrefs

Cf. A006950, A036498, A049453, A210964, A247223, A263536, A308399.

Programs

  • Mathematica
    nmax = 78; CoefficientList[Series[1/Sum[(-x)^(k (6 k + 1)), {k, -nmax, nmax}], {x, 0, nmax}], x]
nmax = 78; CoefficientList[Series[Product[1/((1 - x^(12 k - 7)) * (1 - x^(12 k - 5)) * (1 - x^(12 k))), {k, 1, nmax}], {x, 0, nmax}], x]

Formula

G.f.: 1 / Sum_{k>=1} (-x)^A036498(k).
G.f.: Product_{k>=1} 1 / ((1 - x^(12*k - 7)) * (1 - x^(12*k - 5)) * (1 - x^(12*k))).
a(n) ~ (sqrt(3) - 1) * exp(sqrt(n/6)*Pi) / (2^(5/2)*n). - Vaclav Kotesovec, May 25 2019
Showing 1-4 of 4 results.

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