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.

A260215 Expansion of chi(-q) * chi(q^9) / (chi(q) * chi(-q^9)) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -2, 2, -4, 6, -8, 12, -16, 22, -28, 36, -48, 60, -76, 96, -120, 150, -184, 228, -280, 340, -416, 504, -608, 732, -878, 1052, -1252, 1488, -1768, 2088, -2464, 2902, -3408, 3996, -4672, 5460, -6364, 7400, -8600, 9972, -11544, 13344, -15400, 17752, -20424
Offset: 0

Views

Author

Michael Somos, Aug 13 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 2*x^2 - 4*x^3 + 6*x^4 - 8*x^5 + 12*x^6 - 16*x^7 + 22*x^8 + ...

Links

Crossrefs

Cf. A128143, A260057, A261154, A261156, A261203, A261230, A261325.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q, q^2] QPochhammer[ q, -q] QPochhammer[ -q^9, q^18] QPochhammer[ -q^9, q^9], {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^18 + A)^3 / (eta(x^2 + A)^3 * eta(x^9 + A)^2 * eta(x^36 + A)), n))};

Formula

Expansion of psi(-q) * psi(q^9) / (psi(q) * psi(-q^9)) in powers of q where psi() is a Ramanujan theta function.
Expansion of eta(q)^2 * eta(q^4) * eta(q^18)^3 / (eta(q^2)^3 * eta(q^9)^2 * eta(q^36)) in powers of q.
Euler transform of period 36 sequence [ -2, 1, -2, 0, -2, 1, -2, 0, 0, 1, -2, 0, -2, 1, -2, 0, -2, 0, -2, 0, -2, 1, -2, 0, -2, 1, 0, 0, -2, 1, -2, 0, -2, 1, -2, 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. of A128143.
a(n) = (-1)^n * A261156(n). Convolution inverse of A261156
a(2*n + 1) = -2 * A261203(n) = -2 * A261154(2*n + 1). 2 * a(2*n) = A261154(2*n) unless n=0.
a(3*n) = A261320(n). a(3*n + 1) = -2 * A261325(n). a(3*n + 2) = 2 * A260057(n). - Michael Somos, Nov 08 2015
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A128144 Expansion of chi(-q)* chi(-q^2)* chi(-q^9)/( chi(-q^3)* chi(q^9)) in powers of q where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, -1, 1, 0, -1, 0, 1, 1, -2, 0, 3, 0, -2, 0, 3, 0, -5, 0, 4, -2, -4, 0, 5, 0, -7, 2, 7, 0, -5, 0, 10, 1, -12, 0, 10, 0, -14, -4, 17, 0, -21, 0, 22, 4, -24, 0, 34, 0, -33, 1, 36, 0, -45, 0, 45, -8, -52, 0, 55, 0, -62, 8, 71, 0, -70, 0, 88, 2, -96, 0, 98, 0, -122, -14, 133, 0, -148, 0, 163, 14, -182, 0, 217, 0, -216
Offset: 0

Views

Author

Michael Somos, Feb 16 2007

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

Programs

  • Mathematica
    A128144[n_] := SeriesCoefficient[((QPochhammer[q]*QPochhammer[q^6] *QPochhammer[q^36]*QPochhammer[q^9]^2)/(QPochhammer[q^3]*QPochhammer[q^4] *QPochhammer[q^18]^3)), {q, 0, n}]; Table[A128144[n], {n, 0, 50}] (* G. C. Greubel, Oct 09 2017 *)
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x+A)*eta(x^6+A)*eta(x^36+A)*eta(x^9+A)^2/ (eta(x^3+A)*eta(x^4+A)*eta(x^18+A)^3), n))}

Formula

Expansion of (eta(q)* eta(q^6)* eta(q^36)* eta(q^9)^2)/(eta(q^3)* eta(q^4)* eta(q^18)^3) in powers of q.
Euler transform of period 36 sequence [ -1, -1, 0, 0, -1, -1, -1, 0, -2, -1, -1, 0, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, 0, -1, -1, -2, 0, -1, -1, -1, 0, 0, -1, -1, 0, ...].
G.f. A(x) satisfies 0=f(A(x), A(x^2)) where f(u, v)= (1-v)*(1-v+v^2)*(2*u-u^2)^2 -(u+v-u*v)^2*(u-v)^2.
a(6*n+4)=0. a(6*n)=0 if n>0.
A092848(n) = -a(6*n+2).
A128143(n) = -a(n) if n>0.
A128145(n) = -a(n) if n>0.

A128145 Expansion of psi(q^3)* phi(-q^3)* chi^2(-q^3)/( psi(-q)* phi(-q^18)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, -1, 0, 1, 0, -1, -1, 2, 0, -3, 0, 2, 0, -3, 0, 5, 0, -4, 2, 4, 0, -5, 0, 7, -2, -7, 0, 5, 0, -10, -1, 12, 0, -10, 0, 14, 4, -17, 0, 21, 0, -22, -4, 24, 0, -34, 0, 33, -1, -36, 0, 45, 0, -45, 8, 52, 0, -55, 0, 62, -8, -71, 0, 70, 0, -88, -2, 96, 0, -98, 0, 122, 14, -133, 0, 148, 0, -163, -14, 182, 0, -217, 0, 216
Offset: 0

Views

Author

Michael Somos, Feb 16 2007

Keywords

Comments

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

Links

Crossrefs

A092848(n) = a(6n+2). A128143(n) = a(n) if n > 0. A128144(n) = -a(n) if n > 0.

Programs

  • Mathematica
    eta[x_] := x^(1/24)*QPochhammer[x]; A128145[n_] := SeriesCoefficient[ eta[q^2]*eta[q^3]^3*eta[q^36]/(eta[q]*eta[q^4]*eta[q^6]*eta[q^18]^2 ), {q, 0, n}]; Table[A128145[n], {n,0,50}] (* G. C. Greubel, Aug 16 2017 *)
  • PARI
    {a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)*eta(x^36+A)*eta(x^3+A)^3/ (eta(x+A)*eta(x^4+A)*eta(x^6+A)*eta(x^18+A)^2), n))}

Formula

Expansion of (eta(q^2)* eta(q^3)^3* eta(q^36))/(eta(q)* eta(q^4)* eta(q^6)* eta(q^18)^2) in powers of q.
Euler transform of period 36 sequence [ 1, 0, -2, 1, 1, -2, 1, 1, -2, 0, 1, -1, 1, 0, -2, 1, 1, 0, 1, 1, -2, 0, 1, -1, 1, 0, -2, 1, 1, -2, 1, 1, -2, 0, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v - 1)*(3 - 3*v + v^2)*(2*u - u^2)^2 - (u + v - u*v)^2*(u - v)^2.
a(6n+4)=0. a(6n)=0 if n > 0.

A233034 Expansion of (f(-x^2) / phi(-x^3))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, -2, 4, -1, -8, 14, -4, -23, 40, -10, -60, 98, -24, -140, 224, -54, -304, 478, -112, -627, 968, -224, -1236, 1884, -432, -2346, 3540, -801, -4320, 6454, -1448, -7742, 11472, -2556, -13548, 19936, -4408, -23226, 33952, -7462, -39080, 56800, -12416, -64660
Offset: 0

Views

Author

Michael Somos, Dec 03 2013

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 - 2*x^2 + 4*x^3 - x^4 - 8*x^5 + 14*x^6 - 4*x^7 - 23*x^8 + 40*x^9 + ...
G.f. = q - 2*q^13 + 4*q^19 - q^25 - 8*q^31 + 14*q^37 - 4*q^43 - 23*q^49 + ...

Links

Crossrefs

Cf. A058531, A062242, A062244, A092848, A093073, A128111, A128143, A128144, A128145, A132976, A139032, A143840, A164268, A164612, A164615, A182032, A182033, A182034, A182035, A182056, A182057, A193261.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^2] QPochhammer[ x^6] / QPochhammer[ x^3]^2)^2, {x, 0, n}];
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A) / eta(x^3 + A)^2)^2, n))};

Formula

Expansion of q^(-2/3) * b(q^2) * c(q^2) / (3 * f(-q^3)^4) in powers of q where b(), c() are cubic AGM theta functions.
Expansion of q^(-1/6) * (eta(q^2) * eta(q^6) / eta(q^3)^2)^2 in powers of q.
Euler transform of period 6 sequence [ 0, -2, 4, -2, 0, 0, ...].
G.f.: Product_{k>0} ( (1 - x^(2*k)) * (1 - x^(6*k)) / (1 - x^(3*k))^2 )^2.
a(n) = A092848(2*n) = A128111(2*n) = A182057(4*n) = A062242(4*n + 1) = A182056(4*n + 1) = A139032(6*n + 1) = A164615(6*n + 1) = A182033(6*n + 1) = A058531(12*n + 2) = A093073(12*n + 2) = A128143(12*n + 2) = A128145(12*n + 2) = A143840(12*n + 2) = A182032(12*n + 2) = A193261(12*n + 2).
-a(n) = A062244(4*n + 1) = A182034(6*n + 1) = A182035(6*n + 1) = A128144(12*n + 2) = A132976(12*n + 3) = A164268(12*n + 2) = A164612(12*n + 3) = A182035(12*n + 2).
Showing 1-4 of 4 results.

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