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.

A233698 Expansion of b(q^2) * c(q^2) / (3 * b(q)^2) in powers of q where b(), c() are cubic AGM functions.

Original entry on oeis.org

1, 6, 25, 84, 248, 666, 1662, 3912, 8774, 18894, 39289, 79248, 155612, 298338, 559812, 1030224, 1862647, 3313494, 5807096, 10037796, 17129888, 28886052, 48170178, 79492824, 129900206, 210314976, 337545438, 537278124, 848509124, 1330069554, 2070183912
Offset: 0

Views

Author

Michael Somos, Dec 14 2013

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 6*x + 25*x^2 + 84*x^3 + 248*x^4 + 666*x^5 + 1662*x^6 + 3912*x^7 + ...
G.f. = q^2 + 6*q^5 + 25*q^8 + 84*q^11 + 248*q^14 + 666*q^17 + 1662*q^20 + ...

Links

Crossrefs

Cf. A132977, A164271, A233670.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[((1-x^(2*k)) * (1-x^(3*k)) * (1-x^(6*k)) / (1-x^k)^3)^2,{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-2/3) *(eta[q^2]*eta[q^3]*eta[q^6]/eta[q]^3)^2, {q, 0, 50}], q] (* G. C. Greubel, Aug 07 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A) / eta(x + A)^3)^2, n))}

Formula

Expansion of (eta(q^2) * eta(q^3) * eta(q^6) / eta(q)^3)^2 in powers of q.
Euler transform of period 6 sequence [ 6, 4, 4, 4, 6, 0, ...].
a(n) = (-1)^n * A164271(n). 2 * a(n) = A132977(2*n + 1). -3 * a(n) = A233670(6*n + 4).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A164269 Expansion of q * f(q^9)^3 * phi(q) / (f(q^3) * phi(q^3)^3) in powers of q where f(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, -7, -12, 0, 32, 50, 0, -114, -168, 0, 350, 496, 0, -967, -1332, 0, 2468, 3324, 0, -5916, -7824, 0, 13471, 17548, 0, -29384, -37788, 0, 61784, 78578, 0, -125838, -158496, 0, 249230, 311224, 0, -481506, -596676, 0, 909788, 1119624, 0, -1684824, -2060448, 0
Offset: 1

Views

Author

Michael Somos, Aug 11 2009

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 - 7*q^4 - 12*q^5 + 32*q^7 + 50*q^8 - 114*q^10 - 168*q^11 + ...

Links

Crossrefs

Cf. A164268, A164270, A164271.

Programs

  • Mathematica
    f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A164269[n_] := SeriesCoefficient[q*(f[q^9, -q^18]^3*f[q, q])/(( f[q^3, -q^6])*f[q^3, q^3]^3), {q, 0, n}]; Rest[Table[A164269[n], {n,0,50}]] (* G. C. Greubel, Sep 16 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^9]^3 EllipticTheta[ 3, 0, x] / (QPochhammer[ -x^3] EllipticTheta[ 3, 0, x^3]^3), {x, 0, n}]; (* Michael Somos, Sep 17 2017 *)
a[ n_] := SeriesCoefficient[ x QPochhammer[ -x^3, x^6] QPochhammer[ x^9]^3 EllipticTheta[ 3, 0, x] / EllipticTheta[ 3, 0, x^3]^4, {x, 0, n}]; /(* Michael Somos, Sep 20 2017 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^7 * eta(x^12 + A)^7 * eta(x^18 + A)^9 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A)^18 * eta(x^9 + A)^3 * eta(x^36 + A)^3), n))};

Formula

Euler transform of period 36 sequence [2, -3, -5, -1, 2, 8, 2, -1, -2, -3, 2, 3, 2, -3, -5, -1, 2, 2, 2, -1, -5, -3, 2, 3, 2, -3, -2, -1, 2, 8, 2, -1, -5, -3, 2, 0, ...].
a(3*n) = 0. a(3*n + 1) = A164270(n). a(3*n + 2) = 2 * A164271(n).
Convolution inverse of A164268.
Expansion of x * phi(x) * chi(x^3) * f(x^9)^3 / phi(x^3)^4 = x * phi(x) * f(x^9)^3 / (chi(x^3)^3 * f(x^3)^4) in powers of x where phi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Sep 20 2017

A294387 Expansion of chi(q^3) / chi^3(q) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -3, 6, -12, 21, -36, 60, -96, 150, -228, 342, -504, 732, -1050, 1488, -2088, 2901, -3996, 5460, -7404, 9972, -13344, 17748, -23472, 30876, -40413, 52644, -68268, 88152, -113364, 145224, -185352, 235734, -298800, 377514, -475488, 597108, -747690, 933672
Offset: 0

Views

Author

Michael Somos, Oct 29 2017

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 - 3*x + 6*x^2 - 12*x^3 + 21*x^4 - 36*x^5 + 60*x^6 - 96*x^7 + ...

Links

Crossrefs

Cf. A128111, A128128, A132972, A164270, A164271.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q, -q]^3 / QPochhammer[ q^3, -q^3], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^4 + A)^3 * eta(x^6 + A)^2 / (eta(x^2 + A)^6 * eta(x^3 + A) * eta(x^12 + A)), n))};
  • PARI
    lista(nn) = {q='q+O('q^nn); Vec(eta(q)^3*eta(q^4)^3*eta(q^6)^2/(eta(q^2)^6*eta(q^3)*eta(q^12)))} \\ Altug Alkan, Mar 21 2018

Formula

Expansion of eta(q)^3 * eta(q^4)^3 * eta(q^6)^2 / (eta(q^2)^6 * eta(q^3) * eta(q^12)) in powers of q.
Expansion of (c(q) - c(q^4)) * (c(q) - 4*c(q^4)) / (c(q) + 2*c(q^4))^2 in powers of q where c(q) is a cubic AGM theta function.
Expansion of b(q^2) / b(-q) = b(q^2) / (2*b(q^4) - b(q)) in powers of q where b() is a cubic AGM theta function.
Expansion of (3*a(q^12) - a(q^4)) / (a(q) + a(q^2)) = -1/2 + 3/2*(a(-q^3) + 2*a(q^3)) / (2*a(q) + a(-q)) in powers of q where a() is a cubic AGM theta function.
Euler transform of period 12 sequence [-3, 3, -2, 0, -3, 2, -3, 0, -2, 3, -3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128111.
G.f. A(q) = (1 - T(q)) / (1 + 2*T(q)) where T(q) = q*A128111(q^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u*v) + 3*(u*v)^2 - 4*(u*v)^3 + 2*(u*v)^4 - (u^3 + v^3).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u*(1 + u + u^2) - v^3*(1 - 2*u + 4*u^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) = u1 + u2 + u1*u2 - u3*u6 - 2*u1*u2*u3*u6.
G.f.: Product_{k>0} (1 + x^(6*k-3)) / (1 + x^(2*k-1))^3.
a(n) = (-1)^n * A128128(n). Convolution inverse of A132972.
a(3*n + 1) = -3 * A164270(n). a(3*n + 2) = 6 * A164271(n).
Empirical : Sum_{n>=0} a(n)/exp(Pi*n) = 1/2*(2+2*3^(1/2))^(1/3), validated up to 1000 digits. - Simon Plouffe, May 06 2023
Showing 1-3 of 3 results.

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