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 1-7 of 7 results.

A132975 Expansion of q * psi(-q^9) / psi(-q) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548
Offset: 1

Views

Author

Michael Somos, Sep 07 2007

Keywords

Comments

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

Examples

			G.f. = q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A128129, A128640, A132302, A132972, A132976. Essentially the same as A213267.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1+x^k) * (1-x^(9*k)) * (1+x^(18*k)) / (1-x^(4*k)),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, q^(9/2)] / EllipticTheta[ 2, Pi/4, q^(1/2)], {q, 0, n}]; (* Michael Somos, Oct 31 2015 *)
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^2) * eta(q^9) * eta(q^36) / (eta(q) * eta(q^4) * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 1, 0, ...].
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 - (1 + u1 + u2) * (u3 + u6 + 3 * u3 * u6).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132976.
G.f.: x * Product_{k>0} P(3,x^k) * P(9,x^k) * P(12,x^k) * P(36,x^k) where P(n,x) is the n-th cyclotomic polynomial.
3 * a(n) = A132972(n) unless n=0. a(2*n) = A128129(n). a(2*n + 1) = A132302(n). a(3*n) = A128640(n). Convolution inverse of A132976.
a(n) ~ exp(2*Pi*sqrt(n)/3) / (6 * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015

A132301 Expansion of f(-x, -x^5) * f(-x)^2 / f(-x^6)^3 in powers of x where f(, ) and f() are Ramanujan theta functions.

Original entry on oeis.org

1, -3, 1, 3, -1, 0, 1, -6, 0, 6, -3, 3, 4, -12, 1, 12, -6, 3, 5, -24, 1, 24, -10, 6, 11, -42, 4, 42, -19, 12, 17, -72, 4, 69, -31, 18, 31, -120, 9, 114, -50, 30, 46, -189, 11, 180, -79, 48, 77, -294, 21, 276, -122, 72, 112, -450, 28, 420, -183, 108, 173, -672
Offset: 0

Views

Author

Michael Somos, Aug 17 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 + x^2 + 3*x^3 - x^4 + x^6 - 6*x^7 + 6*x^9 - 3*x^10 + 3*x^11 + ...
G.f. = 1/q - 3*q^2 + q^5 + 3*q^8 - q^11 + q^17 - 6*q^20 + 6*q^26 - 3*q^29 + ...

Links

Crossrefs

Cf. A092848, A132179, A132302.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^6] QPochhammer[ x^5, x^6] QPochhammer[ x]^2 / QPochhammer[ x^6]^2, {x, 0, n}]; (* Michael Somos, Nov 01 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^6 + A)), n))};

Formula

Expansion of q^(1/3) * eta(q)^3 / (eta(q^2) * eta(q^3) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ -3, -2, -2, -2, -3, 0, ...].
Given g.f. A(x), then B(q) = A(q^3) / (3*q) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132302.
a(2*n) = A132179(n). a(2*n + 1) = -3 * A092848(n). - Michael Somos, Nov 01 2015

A141094 Expansion of b(q) / b(q^2) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, -3, 3, -3, 6, -9, 12, -15, 21, -30, 36, -45, 60, -78, 96, -117, 150, -189, 228, -276, 342, -420, 504, -603, 732, -885, 1050, -1245, 1488, -1773, 2088, -2454, 2901, -3420, 3996, -4662, 5460, -6378, 7404, -8583, 9972, -11565, 13344, -15378, 17748, -20448
Offset: 0

Views

Author

Michael Somos, Jun 04 2008, Aug 12 2009

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).
For n >= 1, a(n)/3 is a weighted count of overpartitions with restricted odd differences. Namely, the number of overpartitions of n counted with weight (-1)^(the largest part) and such that: (i) the difference between successive parts may be odd only if the larger part is overlined and (ii) the smallest part of the overpartition is odd and overlined. - Jeremy Lovejoy, Aug 07 2015

Examples

			G.f. = 1 - 3*q + 3*q^2 - 3*q^3 + 6*q^4 - 9*q^5 + 12*q^6 - 15*q^7 + 21*q^8 + ...

Links

Crossrefs

Cf. A092848, A124243, A128128, A132302.

Programs

  • Maple
    with(numtheory):
a:= proc(n) option remember:
      `if`(n=0, 1, add(add(d*[0, -3, 0, -2, 0, -3]
      [irem(d, 6)+1], d=divisors(j))*a(n-j), j=1..n)/n)
    end:
seq(a(n), n=0..60);  # Alois P. Heinz, Aug 08 2015
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2]^3 QPochhammer[ -x^3, x^3], {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[n_] := a[n] = If[n==0, 1, Sum[Sum[d{0, -3, 0, -2, 0, -3}[[Mod[d, 6]+1]], {d, Divisors[j]}] a[n-j], {j, 1, n}]/n];
a /@ Range[0, 60] (* Jean-François Alcover, Jan 01 2021, after Alois P. Heinz *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 * eta(x^6 + A) / (eta(x^2 + A)^3 * eta(x^3 + A)), n))};

Formula

Expansion of chi(-q)^3 / chi(-q^3) in powers of q where chi() is a Ramanujan theta function.
Expansion of eta(q)^3 * eta(q^6) / (eta(q^2)^3 * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [ -3, 0, -2, 0, -3, 0, ...].
G.f.: Product_{k>0} (1 - x^(2*k-1))^3 / (1 - x^(6*k-3)).
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = v^2 - u * (2 - u*v).
G.f. A(x) satisfies 0 = f(A(x), A(x^3)) where f(u, v) = u * (u^2 - 2*u + 4) - v^3 * (u^2 + u + 1).
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 * (u6^2 - u2 * u3) - u6 * (u3^2 - u6*u2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A092848.
a(n) = -3 * A124243(n) unless n=0. a(n) = (-1)^n * A132972(n).
a(2*n) = A128128(n). a(2*n + 1) = - 3* A132302(n).
Convolution inverse of A128128.
Empirical: Sum_{n>=1} exp(-Pi)^(n-1)*(-1)^(n+1)*a(n) = (-2+2*3^(1/2))^(1/3). - Simon Plouffe, Feb 20 2011
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n)/3) / (2*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A132180 Expansion of f(q, q^2) * f(-q^3) / f(-q^2)^2 in powers of q where f(, ), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 1, 6, 3, 12, 5, 21, 10, 36, 15, 60, 26, 96, 39, 150, 63, 228, 92, 342, 140, 504, 201, 732, 295, 1050, 415, 1488, 591, 2088, 818, 2901, 1140, 3996, 1554, 5460, 2126, 7404, 2861, 9972, 3855, 13344, 5126, 17748, 6816, 23472, 8970, 30876, 11793, 40413
Offset: 0

Views

Author

Michael Somos, Aug 12 2007

Keywords

Comments

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

Examples

			G.f. = 1 + q + 3*q^2 + q^3 + 6*q^4 + 3*q^5 + 12*q^6 + 5*q^7 + 21*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A128128, A132179, A132302.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^3]^3 / (QPochhammer[ q] QPochhammer[ q^2] QPochhammer[ q^6]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -q, q^3] QPochhammer[ -q^2, q^3] QPochhammer[ q^3]^2 / QPochhammer[ q^2]^2, {q, 0, n}]; (* Michael Somos, Nov 01 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A)^3 / (eta(x + A) * eta(x^2 + A) * eta(x^6 + A)), n))};

Formula

Expansion of eta(q^3)^3 / (eta(q) * eta(q^2) * eta(q^6)) in powers of q.
Euler transform of period 6 sequence [ 1, 2, -2, 2, 1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (v^2 - 2*u)^3 - u^4 * (2*u - 3*v^2) * (4*u - 3*v^2).
G.f. is a period 1 Fourier series which satisfies f(-1 / (6 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132179.
G.f.: Product_{k>0} (1 + x^k + x^(2*k))^2 / ( (1 + x^k)^2 * (1 - x^k + x^(2*k))).
a(2*n) = A128128(n). a(2*n + 1) = A132302(n).

A213267 Expansion of phi(q^9) / (psi(-q) * chi(q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 4, 5, 7, 10, 12, 15, 20, 26, 32, 39, 50, 63, 76, 92, 114, 140, 168, 201, 244, 295, 350, 415, 496, 591, 696, 818, 967, 1140, 1332, 1554, 1820, 2126, 2468, 2861, 3324, 3855, 4448, 5126, 5916, 6816, 7824, 8970, 10292, 11793, 13471, 15372, 17548
Offset: 0

Views

Author

Michael Somos, Jun 07 2012

Keywords

Comments

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

Examples

			1 + q + q^2 + q^3 + 2*q^4 + 3*q^5 + 4*q^6 + 5*q^7 + 7*q^8 + 10*q^9 + ...

Links

Crossrefs

Cf. A128129, A132302, A132975, A132977, A132978, A164617.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(6*k)) * (1+x^(9*k))^5 * (1-x^(9*k))^3 / ((1-x^(4*k)) * (1+x^(3*k)) * (1-x^(36*k))^2),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 14 2015 *)
  • 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^12 + A) * eta(x^18 + A)^5 / (eta(x + A) * eta(x^4 + A) * eta(x^6 + A)^2 * eta(x^9 + A)^2 * eta(x^36 + A)^2), n))}

Formula

Expansion of eta(q^2) * eta(q^3) * eta(q^12) * eta(q^18)^5 / (eta(q) * eta(q^4) * eta(q^6)^2 * eta(q^9)^2 * eta(q^36)^2) in powers of q.
Euler transform of period 36 sequence [ 1, 0, 0, 1, 1, 1, 1, 1, 2, 0, 1, 1, 1, 0, 0, 1, 1, -2, 1, 1, 0, 0, 1, 1, 1, 0, 2, 1, 1, 1, 1, 1, 0, 0, 1, 0, ...].
a(n) = A132975(n) unless n=0.
a(2*n) = A128129(n). a(2*n + 1) = A132302.
a(3*n) = A164617(n). a(3*n + 1) = A132977(n). a(3*n + 2) = A132978(n).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (2 * 3^(3/2) * n^(3/4)). - Vaclav Kotesovec, Oct 14 2015

A254346 Expansion of f(x, x^5) * f(-x^6) / f(x)^2 in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 3, -5, 10, -15, 26, -39, 63, -92, 140, -201, 295, -415, 591, -818, 1140, -1554, 2126, -2861, 3855, -5126, 6816, -8970, 11793, -15372, 20007, -25857, 33356, -42771, 54734, -69683, 88530, -111968, 141312, -177642, 222842, -278557, 347484, -432095, 536230
Offset: 0

Views

Author

Michael Somos, Jan 29 2015

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 - 5*x^3 + 10*x^4 - 15*x^5 + 26*x^6 - 39*x^7 + ...
G.f. = q - q^3 + 3*q^5 - 5*q^7 + 10*q^9 - 15*q^11 + 26*q^13 - 39*q^15 + ...

Links

Crossrefs

Cf. A132302, A254372.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^3] QPochhammer[ x^12] / (QPochhammer[ x^2] QPochhammer[ -x]), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / eta(x^2 + A)^4, n))};

Formula

Expansion of q^(-1/2) * eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / eta(q^2)^4 in powers of q.
Euler transform of period 12 sequence [ -1, 3, -2, 2, -1, 2, -1, 2, -2, 3, -1, 0, ...].
a(n) = (-1)^n * A132302(n). 2 * a(n) = A254372(2*n + 1).

A145977 Expansion of c(q^3) / (c(q^3) + c(q^6)) where c() is a cubic AGM function.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 4, -5, 7, -10, 12, -15, 20, -26, 32, -39, 50, -63, 76, -92, 114, -140, 168, -201, 244, -295, 350, -415, 496, -591, 696, -818, 967, -1140, 1332, -1554, 1820, -2126, 2468, -2861, 3324, -3855, 4448, -5126, 5916, -6816, 7824, -8970, 10292, -11793, 13471, -15372, 17548, -20007
Offset: 0

Views

Author

Michael Somos, Oct 26 2008

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 - q + q^2 - q^3 + 2*q^4 - 3*q^5 + 4*q^6 - 5*q^7 + 7*q^8 - 10*q^9 + ...

Links

Crossrefs

Cf. A124243, A128129, A128641, A132302, A139032.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1 - EllipticTheta[ 2, 0, x^(9/2)] / EllipticTheta[ 2, 0, x^(1/2)], {x, 0, n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A) * eta(x^9 + A)^2 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^18 + A)), n))};

Formula

Expansion of 1 - q * psi(q^9) / psi(q) = phi(-q^9) / (psi(q) * chi(-q^3)) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of eta(q) * eta(q^6) * eta(q^9)^2 / (eta(q^2)^2 * eta(q^3) * eta(q^18)), in powers of q.
Euler transform of period 18 sequence [ -1, 1, 0, 1, -1, 1, -1, 1, -2, 1, -1, 1, -1, 1, 0, 1, -1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (18 t)) = (2/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A139032.
G.f.: Product_{k>0} (P(3, x^k) * P(9, x^k)) / (P(4, x^k)^2 * P(18, x^k)) where P(n, x) is the n-th cyclotomic polynomial.
Convolution inverse of A139032.
a(n) = - A124243(n) unless n=0. a(2*n) = A128129(n) = a(2*n) unless n=0.
a(2*n + 1) = - A132302(n). a(3*n) = A128641(n).
Showing 1-7 of 7 results.

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