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.

A213625 Expansion of psi(x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 3, 6, 4, 4, 7, 2, 8, 10, 4, 10, 9, 6, 8, 10, 4, 8, 16, 8, 9, 12, 8, 12, 20, 6, 8, 10, 8, 18, 11, 12, 8, 20, 12, 8, 20, 6, 20, 26, 8, 8, 15, 10, 16, 18, 12, 16, 20, 10, 16, 16, 8, 24, 24, 8, 21, 26, 8, 20, 20, 14, 8, 28, 16, 10, 28, 10, 24, 22, 8, 16, 17
Offset: 0

Views

Author

Michael Somos, Jun 16 2012

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + 3*x^2 + 6*x^3 + 4*x^4 + 4*x^5 + 7*x^6 + 2*x^7 + 8*x^8 + 10*x^9 + ...
G.f. = q + 2*q^5 + 3*q^9 + 6*q^13 + 4*q^17 + 4*q^21 + 7*q^25 + 2*q^29 + 8*q^33 + ...

Links

Crossrefs

Cf. A116597, A132969, A213622.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 3, 0, x^2] / (4 x^(1/4)), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^5 / (eta(x + A)^2 * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(-1/4) * eta(q^2)^2 * eta(q^4)^5 / (eta(q)^2 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ 2, 0, 2, -5, 2, 0, 2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (32 t)) = 2^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A116597.
a(2*n) = A213622(n). a(2*n + 1) = 2 * A132969(n).

A132970 Expansion of phi(-x) * chi(-x) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 2, -1, 5, -5, 3, -5, 6, -10, 10, -8, 13, -15, 15, -16, 23, -27, 25, -30, 35, -40, 42, -45, 55, -66, 68, -70, 86, -95, 100, -110, 125, -141, 150, -161, 185, -207, 215, -235, 266, -293, 310, -335, 375, -410, 438, -470, 521, -575, 610, -653, 725, -785, 835, -900, 983, -1070, 1140, -1220, 1331
Offset: 0

Views

Author

Michael Somos, Sep 04 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 3*x + 2*x^2 - x^3 + 5*x^4 - 5*x^5 + 3*x^6 - 5*x^7 + 6*x^8 + ...
G.f. = 1/q - 3*q^23 + 2*q^47 - q^71 + 5*q^95 - 5*q^119 + 3*q^143 - 5*q^167 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 60, Eqs. (26.64),(26.65),(26.66)

Links

Crossrefs

CF. A124226, A132969.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Jul 20 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, (n+1)\2, 1 - x^(2*k-1), 1 + x * O(x^n)) * sum(k=1, sqrtint(n), 2 * (-1)^k * x^k^2, 1), n))};
  • PARI
    {a(n) = my(A) ; if( n<0, 0, A = x * O(x^n) ; polcoeff( eta(x + A)^3 / eta(x^2 + A)^2, n))};

Formula

Expansion of phi(-q) + 2 * psi(-q) in powers of q where phi(), psi() are Ramanujan 3rd order mock theta functions.
Expansion of q^(1/24) * eta(q)^3 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [ -3, -1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 48^(1/2) (t/i)^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A085140.
G.f.: ( Sum_{k in Z} (-1)^k * x^k^2 ) / ( Product_{k>0} (1 + x^k) ).
G.f.: Product_{k>0} (1 - x^k) / (1 + x^k)^2.
a(n) = (-1)^n * A132969(n). a(n) = A124226(n) unless n=1.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/6)) / (2*sqrt(n)). - Vaclav Kotesovec, Oct 14 2017

A246835 Expansion of psi(-x)^2 * phi(x^2) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 3, -6, 4, -4, 7, -2, 8, -10, 4, -10, 9, -6, 8, -10, 4, -8, 16, -8, 9, -12, 8, -12, 20, -6, 8, -10, 8, -18, 11, -12, 8, -20, 12, -8, 20, -6, 20, -26, 8, -8, 15, -10, 16, -18, 12, -16, 20, -10, 16, -16, 8, -24, 24, -8, 21, -26, 8, -20, 20, -14, 8, -28
Offset: 0

Views

Author

Michael Somos, Sep 04 2014

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 3*x^2 - 6*x^3 + 4*x^4 - 4*x^5 + 7*x^6 - 2*x^7 + 8*x^8 + ...
G.f. = q - 2*q^5 + 3*q^9 - 6*q^13 + 4*q^17 - 4*q^21 + 7*q^25 - 2*q^29 + ...

Links

Crossrefs

Cf. A132969, A213622, A213625, A246836.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3, 0, q^2]* EllipticTheta[2, 0, I*q^(1/2)]^2/(4*(-q)^(1/4)), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Nov 29 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^7 / (eta(x^2 + A)^4 * eta(x^8 + A)^2), n))};

Formula

Expansion of q^(-1/4) * eta(q)^2 * eta(q^4)^7 / (eta(q^2)^4 * eta(q^8)^2) in powers of q.
Euler transform of period 8 sequence [ -2, 2, -2, -5, -2, 2, -2, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 16 (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A246836.
a(n) = (-1)^n * A213625(n). a(2*n) = A213622(n). a(2*n + 1) = -2 * A132969(n).

A260460 Expansion of f(-q) in powers of q where f() is a 3rd order mock theta function.

Original entry on oeis.org

1, -1, -2, -3, -3, -3, -5, -7, -6, -6, -10, -12, -11, -13, -17, -20, -21, -21, -27, -34, -33, -36, -46, -51, -53, -58, -68, -78, -82, -89, -104, -118, -123, -131, -154, -171, -179, -197, -221, -245, -262, -279, -314, -349, -369, -398, -446, -486, -515, -557
Offset: 0

Views

Author

Michael Somos, Jul 26 2015

Keywords

Examples

			G.f. = 1 - x - 2*x^2 - 3*x^3 - 3*x^4 - 3*x^5 - 5*x^6 - 7*x^7 - 6*x^8 + ...

Crossrefs

Cf. A000025, A053250, A053251, A132969.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, SeriesCoefficient[ Sum[ (-x)^k^2 / Product[ 1 + (-x)^j, {j, k}]^2, {k, 0, Sqrt@n}], {x, 0, n}]];
  • PARI
    {a(n) = my(t); if( n<0, 0, t = 1 + O(x^n); polcoeff( sum(k=1, sqrtint(n), t *= (-x)^(2*k - 1) / (1 + (-x)^k)^2 + x * O(x^(n - (k-1)^2)), 1), n))};

Formula

G.f.: Sum_{k>=0} (-x)^(k^2) / Product_{i=1..k} (1 + (-x)^i)^2.
G.f.: 2 * (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2) / (1 + x^k)) / (Sum_{k in Z} (-1)^k * x^(k*(3*k + 1)/2))
a(n) = (-1)^n * A000025(n). a(n) < 0 if n>0.
a(n) = A053250(n) - 2 * A053251(n) = 2 * A053250(n) - A132969(n) = A132969(n) - 4 * A053251(n).
Showing 1-4 of 4 results.

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