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.

A145722 Expansion of f(q) * f(q^5) / phi(-q^2)^2 in powers of q where f(), phi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 4, 8, 12, 21, 30, 48, 68, 102, 143, 207, 284, 400, 542, 744, 996, 1344, 1776, 2361, 3088, 4050, 5248, 6808, 8742, 11232, 14310, 18224, 23052, 29133, 36601, 45936, 57360, 71528, 88812, 110110, 135990, 167704, 206108, 252912, 309408
Offset: 0

Views

Author

Michael Somos, Oct 23 2008

Keywords

Comments

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

Examples

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 8*x^4 + 12*x^5 + 21*x^6 + 30*x^7 + 48*x^8 + ...
G.f. = q + q^5 + 3*q^9 + 4*q^13 + 8*q^17 + 12*q^21 + 21*q^25 + 30*q^29 + ...

Links

Crossrefs

Cf. A036026.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -x]  QPochhammer[ -x^5] / EllipticTheta[ 4, 0, x^2]^2, {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^(2*k)) * (1-x^(10*k)) * (1+x^(5*k)) / ((1-x^k) * (1 + x^(10*k))),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A) * eta(x^10 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q^(-1/4) * eta(q^4) * eta(q^10)^3 / (eta(q) * eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ 1, 2, 1, 1, 2, 2, 1, 1, 1, 0, 1, 1, 1, 2, 2, 1, 1, 2, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 20^(-1/2) g(t), where q = exp(2 Pi i t) and g() is the g.f. for A145723.
G.f.: Product_{k>0} (1 + x^(2*k)) * (1 - x^(10*k)) * (1 + x^(5*k)) / ((1 - x^k) * (1 + x^(10*k))).
a(n) = A036026(2*n).
a(n) ~ exp(2*Pi*sqrt(n/5)) / (4 * 5^(3/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A116494 Expansion of psi(q^5)/psi(q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -2, 3, -3, 4, -6, 8, -10, 12, -16, 21, -25, 30, -38, 48, -57, 68, -84, 102, -121, 143, -172, 207, -243, 284, -338, 400, -465, 542, -636, 744, -862, 996, -1158, 1344, -1546, 1776, -2050, 2361, -2701, 3088, -3540, 4050, -4613, 5248, -5980, 6808, -7719, 8742, -9916, 11232, -12682
Offset: 0

Views

Author

Michael Somos, Feb 18 2006

Keywords

Comments

Ramanujan theta functions: f(q) := Prod_{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) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

Links

Crossrefs

Cf. A036026.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(5/2)]/EllipticTheta[2, 0, q^(1/2)]), {q, 0, n}]; Table[a[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^10+A)^2/eta(x^2+A)^2/eta(x^5+A), n))}

Formula

Expansion of q^(-1/2)*eta(q)*eta(q^10)^2/(eta(q^2)^2*eta(q^5)) in powers of q.
Euler transform of period 10 sequence [ -1,1,-1,1,0,1,-1,1,-1,0,...].
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2)) where f(u,v)=(1-u^2)(1-5u^2)v^2 -(u^2-v^2)^2.
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^4)) where f(u,v,w)=v*w*(1-v^2)-u^2*(v+w)^2.
Given g.f. A(x), then B(x)=x*A(x^2) satisfies 0=f(B(x),B(x^2),B(x^3),B(x^6)) where f(u1,u2,u3,u6)=u2*u6*(u1^2-u3^2) -(u2*u3-u1*u6)^2.
G.f.: Product_{k>0} (1-x^k)/(1-x^(5k))*((1-x^(10k))/(1-x^(2k)))^2 = (Sum_{k>0} x^(5(k^2-k)/2))/(Sum_{k>0} x^((k^2-k)/2)).
a(n) = (-1)^n*A036026(n).

A145708 Expansion of psi(-q) / psi(-q^5) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, -1, 0, 1, 0, 0, -1, 0, 2, 0, 0, -1, 0, 2, -1, 0, -2, 0, 3, -2, 0, -3, 0, 5, -2, 0, -3, 0, 6, -2, 0, -4, 0, 8, -3, 0, -6, 0, 11, -5, 0, -8, 0, 14, -6, 0, -10, 0, 18, -6, 0, -12, 0, 22, -9, 0, -16, 0, 28, -13, 0, -21, 0, 36, -14, 0, -25, 0, 44, -16, 0
Offset: 0

Views

Author

Michael Somos, Oct 17 2008

Keywords

Comments

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

Examples

			G.f. = 1 - x - x^3 + x^5 - x^8 + 2*x^10 - x^13 + 2*x^15 - x^16 - 2*x^18 + ...
G.f. = 1/q - q - q^5 + q^9 - q^15 + 2*q^19 - q^25 + 2*q^29 - q^31 - 2*q^35 + ...

Links

Crossrefs

Cf. A036026, A138532, A145723, A145740, A146164.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ x^(1/2) EllipticTheta[ 2, Pi/4, x^(1/2)] / EllipticTheta[ 2, Pi/4, x^(5/2)], {x, 0, n}]; (* Michael Somos, Sep 06 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^10 + A) / (eta(x^2 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of q^(1/2) * eta(q) * eta(q^4) * eta(q^10) / (eta(q^2) * eta(q^5) * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -1, 0, -1, -1, 0, 0, -1, -1, -1, 0, -1, -1, -1, 0, 0, -1, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = 5^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036026.
a(5*n + 2) = a(5*n + 4) = 0.
G.f.: (Product_{k>0} P(5, x^k) * P(20, x^k))^(-1) where P(n, x) is the n-th cyclotomic polynomial.
a(n) = (-1)^n * A138532(n). a(5*n + 3) = - A036026(n).
Convolution square is A145740. Convolution inverse is A036026.
a(n) = A145723(2*n - 1). a(2*n) = A146164(n). a(2*n + 1) = - A147699(n). - Michael Somos, Sep 06 2015

A147702 Expansion of eta(q) * eta(q^10)^3 / (eta(q^2) * eta(q^4) * eta(q^5) * eta(q^20)) in powers of q.

Original entry on oeis.org

1, -1, 0, -1, 2, -1, 0, -2, 4, -3, 0, -3, 8, -4, 0, -6, 14, -8, 0, -10, 22, -12, 0, -16, 36, -21, 0, -25, 56, -30, 0, -38, 84, -48, 0, -57, 126, -68, 0, -84, 184, -102, 0, -121, 264, -143, 0, -172, 376, -207, 0, -243, 528, -284, 0, -338, 732, -400, 0, -465, 1008, -542, 0, -636, 1374, -744, 0, -862, 1856, -996, 0
Offset: 0

Views

Author

Michael Somos, Nov 10 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^3 + 2*q^4 - q^5 - 2*q^7 + 4*q^8 - 3*q^9 - 3*q^11 + 8*q^12 + ...

Links

Crossrefs

Cf. A036026, A138526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ -q^5] QPochhammer[ q, q^2] / QPochhammer[ q^4], {q, 0, n}]; (* Michael Somos, Sep 02 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^10 + A)^3 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^5 + A) * eta(x^20 + A)), n))};

Formula

Expansion of chi(-q) * f(q^5) / f(-q^4) in powers of q where f(), chi() are Ramanujan theta functions.
Euler transform of period 20 sequence [ -1, 0, -1, 1, 0, 0, -1, 1, -1, -2, -1, 1, -1, 0, 0, 1, -1, 0, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (80 t)) = (5/4)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A147699.
a(2*n + 1) = - A036026(n). a(4*n) = A138526(n). a(4*n + 2) = 0.
Showing 1-4 of 4 results.

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