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 1-5 of 5 results.

A224216 Expansion of q * f(-q,-q^7)^2 / (phi(q^2) * psi(-q)) in powers of q where phi(), psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

1, -1, -2, 3, 4, -6, -8, 11, 15, -20, -26, 34, 44, -56, -72, 91, 114, -143, -178, 220, 272, -334, -408, 498, 605, -732, -884, 1064, 1276, -1528, -1824, 2171, 2580, -3058, -3616, 4269, 5028, -5910, -6936, 8124, 9498, -11088, -12922, 15034, 17468, -20264
Offset: 1

Views

Author

Michael Somos, Apr 01 2013

Keywords

Comments

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

Examples

			q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + 15*q^9 + ...

Links

Crossrefs

Cf. A115671, A208856.

Programs

  • Mathematica
    a[ n_] := (-1)^Floor[ n / 2] SeriesCoefficient[ (QPochhammer[ -q] / QPochhammer[ q] - 1) / 2, {q, 0, n}]
a[ n_] := SeriesCoefficient[ q^(1/2) QPochhammer[ -q] EllipticTheta[ 2, 0, q^2] / EllipticTheta[ 4, 0, q^4]^2 QPochhammer[ q, q^8]^2 QPochhammer[ q^7, q^8]^2 / 2, {q, 0, n}]
  • PARI
    {a(n) = local(A); if( n<1, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (-1 + eta(x^2 + A)^3 / eta(x + A)^2 / eta(x^4 + A)) / 2, n))}

Formula

Expansion of q * (f(-q^8) / f(q^2))^2 * f(-q,-q^7) / f(-q^3,-q^5) = q * f(-q,-q^7) * f(-q^2,-q^6)^2 / (f(-q^3,-q^5) * f(-q^4,-q^4)^2) in powers of q where f() is Ramanujan's theta function.
Euler transform of period 8 sequence [ -1, -2, 1, 4, 1, -2, -1, 0, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2)) where f(u, v) = (u - v) * (1 + 2*u) - (u + u^2) * (1 - 4*v - 4*v^2).
a(n) = (-1)^floor(n \ 2) * A115671(n) unless n=0.

A218171 Expansion of f(x^11, x^13) - x * f(x^5, x^19) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, -1, 0, 0, 0, 0, -1, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 22 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This is an example of the quintuple product identity in the form f(a*b^4, a^2/b) - (a/b) * f(a^4*b, b^2/a) = f(-a*b, -a^2*b^2) * f(-a/b, -b^2) / f(a, b) where a = x^7, b = x. - Michael Somos, Nov 09 2014
This is exercise 0.13 3 on page 45 of Cooper [2017]. - Michael Somos, Aug 30 2025

Examples

			G.f. = 1 - x - x^6 + x^11 + x^13 - x^20 - x^35 + x^46 + x^50 - x^63 - x^88 + ...
G.f. = q - q^49 - q^289 + q^529 + q^625 - q^961 - q^1681 + q^2209 + q^2401 + ...

Links

Crossrefs

Cf. A010815, A115671.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, If[ OddQ[ DivisorSigma[ 0, 48 n + 1]], JacobiSymbol[ 6, Sqrt[48 n + 1]], 0]]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ -q] + QPochhammer[ q]) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q] (QPochhammer[ q^2]^3 / QPochhammer[ q]^2/ QPochhammer[ q^4] + 1) / 2, {q, 0, 2 n}]; (* Michael Somos, Nov 09 2014 *)
  • PARI
    {a(n) = my(m); if( issquare(48*n + 1, &m), kronecker(6, m), 0)};
  • PARI
    {a(n) = my(m); if( n<0, 0, m = 2*n; polcoeff( eta(x + x * O(x^m)), m))};

Formula

Expansion of f(x^3, x^5) * chi(-x) in powers of x where f(, ) is Ramanujan's general theta function and chi() is a Ramanujan theta function.
G.f.: Sum_{k in Z} x^(12*k^2 + k) - x^(12*k^2 + 7*k + 1).
a(n) = A010815(2*n) for all n in Z.
G.f.: Product_{j>0} (1-x^(8*j-1)) * (1-x^(8*j-7)) * (1-x^(8*j)) * (1-x^(16*j-6)) * (1-x^(16*j-10)). [Cooper 2017] - Michael Somos, Aug 30 2025

A185083 Partitions of 2*n into parts not congruent to 0, +-2, +-12, +-14, 16 (mod 32).

Original entry on oeis.org

1, 1, 3, 6, 11, 20, 34, 56, 91, 143, 220, 334, 498, 732, 1064, 1528, 2171, 3058, 4269, 5910, 8124, 11088, 15034, 20264, 27154, 36189, 47988, 63324, 83176, 108780, 141672, 183776, 237499, 305812, 392406, 501856, 639781, 813108, 1030354, 1301928, 1640572
Offset: 0

Views

Author

Michael Somos, Mar 02 2012

Keywords

Comments

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

Examples

			1 + x + 3*x^2 + 6*x^3 + 11*x^4 + 20*x^5 + 34*x^6 + 56*x^7 + 91*x^8 + ...

Links

Crossrefs

Cf. A115671, A208850, A208851.

Programs

  • Mathematica
    f[x_, y_] := QPochhammer[-x, x*y]*QPochhammer[-y, x*y]*QPochhammer[x*y, x*y]; A185083[n_] := SeriesCoefficient[(1/2)*(f[x^2, x^2]/f[-x, -x] + 1), {x, 0, n}]; Table[A185083[n], {n,0,50}] (* G. C. Greubel, Jun 22 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n = 2*n; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) + 1) / 2, n))}

Formula

Expansion of (phi(q^2) / phi(-q) + 1) / 2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 16 sequence [ 1, 2, 3, 2, 3, 0, 1, 0, 1, 0, 3, 2, 3, 2, 1, 0, ...].
2 * a(n) = A208850(n) unless n = 0. a(n + 1) = A208851(n). a(n) = A115671(2*n).

A208856 Partitions of n into parts not congruent to 0, +-4, +-6, +-10, 16 (mod 32).

Original entry on oeis.org

1, 1, 2, 3, 4, 6, 8, 11, 15, 20, 26, 34, 44, 56, 72, 91, 114, 143, 178, 220, 272, 334, 408, 498, 605, 732, 884, 1064, 1276, 1528, 1824, 2171, 2580, 3058, 3616, 4269, 5028, 5910, 6936, 8124, 9498, 11088, 12922, 15034, 17468, 20264, 23472, 27154, 31369, 36189
Offset: 0

Views

Author

Michael Somos, Mar 02 2012

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Andrews (1987) refers to this sequence as p(T, n) where T is the set in equation (1) on page 437.

Examples

			1 + x + 2*x^2 + 3*x^3 + 4*x^4 + 6*x^5 + 8*x^6 + 11*x^7 + 15*x^8 + 20*x^9 + ...
a(5) = 6 since  5 = 3 + 2 = 3 + 1 + 1 = 2 + 2 + 1 = 2 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 in 6 ways.
a(6) = 8 since  5 + 1 = 3 + 3 = 3 + 2 + 1 = 3 + 1 + 1 + 1 = 2 + 2 + 2 = 2 + 2 + 1 + 1 = 2 + 1 + 1 + 1 + 1 = 1 + 1 + 1 + 1 + 1 + 1 in 8 ways.

Links

Crossrefs

Cf. A080054, A115671, A187154, A208851.

Programs

  • Mathematica
    A208856[n_] := SeriesCoefficient[(1/(2*q))*((QPochhammer[-q, -q]/ QPochhammer[q, q]) - 1), {q, 0, n}]; Table[A208856[n], {n,0,50}] (* G. C. Greubel, Jun 19 2017 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, n++; A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A)) - 1) / 2, n))}

Formula

Expansion of (f(x) / f(-x) - 1) / (2 * x) in powers of x where f() is a Ramanujan theta function.
Expansion of (f(x^14, x^34) - x^4 * f(x^2, x^46)) / f(-x, -x^2) in powers of x where f() is Ramanujan's two-variable theta function.
Euler transform of period 32 sequence [ 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 1, 1, 0, 1, 1, 1, 0, 1, 0, 1, 1, 1, 0, ...].
a(n) = A115671(n + 1). 2 * a(n) = A080054(n + 1). a(2*n) = A187154(n). a(2*n + 1) = A208851(n).

A245432 Expansion of f(-q^3, -q^5)^2 / (psi(-q) * phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, -1, -2, 3, 4, -6, -8, 11, 15, -20, -26, 34, 44, -56, -72, 91, 114, -143, -178, 220, 272, -334, -408, 498, 605, -732, -884, 1064, 1276, -1528, -1824, 2171, 2580, -3058, -3616, 4269, 5028, -5910, -6936, 8124, 9498, -11088, -12922, 15034, 17468, -20264
Offset: 0

Views

Author

Michael Somos, Jul 21 2014

Keywords

Comments

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

Examples

			G.f. = 1 + q - q^2 - 2*q^3 + 3*q^4 + 4*q^5 - 6*q^6 - 8*q^7 + 11*q^8 + ...

Links

Crossrefs

Cf. A115671, A210063, A226192, A224216, A244526.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2, q^4] / QPochhammer[ q^4, q^8]^2)^2 QPochhammer[ q^3, q^8] QPochhammer[ q^5, q^8] / (QPochhammer[ q^1, q^8] QPochhammer[ q^7, q^8]), {q, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x * O(x^n))^[0, -1, 2, 1, -4, 1, 2, -1][k%8 + 1]), n))};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); (-1)^(n \ 2) * polcoeff( (1 + eta(x^2 + A)^3 / (eta(x + A)^2 * eta(x^4 + A))) / 2, n))};

Formula

Expansion of (f(-q^3, -q^5) / f(-q^1, -q^7)) * (psi(q^4) / phi(q^2)) in powers of q where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 8 sequence [ 1, -2, -1, 4, -1, -2, 1, 0, ...].
Convolution quotient of A244526 and A226192.
a(n) = (-1)^floor(n/2) * A115671(n).
a(n) = A224216(n) unless n=0. a(2*n+1) = A210063(n).
Showing 1-5 of 5 results.

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