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.

A128591 Expansion of f(x, x^5) * f(x, x^3) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 3, 0, 0, 1, 2, 2, 1, 1, 1, 1, 2, 3, 1, 1, 0, 2, 1, 1, 2, 0, 2, 0, 2, 1, 0, 4, 2, 0, 1, 1, 2, 1, 2, 2, 1, 2, 0, 1, 1, 2, 0, 1, 1, 1, 2, 2, 2, 0, 1, 1, 3, 1, 1, 0, 1, 4, 1, 2, 1, 0, 4, 0, 0, 1, 1, 2, 1, 2, 1, 1, 0, 1, 1, 1, 2, 3
Offset: 0

Views

Author

Michael Somos, Mar 11 2007

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + x^2 + x^3 + x^4 + x^5 + 2*x^6 + x^7 + 2*x^8 + x^9 + x^10 + 3*x^11 + ...
G.f. = q^11 + 2*q^35 + q^59 + q^83 + q^107 + q^131 + 2*q^155 + q^179 + 2*q^203 + ...

Links

Crossrefs

Cf. A128580, A128582, A190615, A257920, A257921,A260118.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/2) QPochhammer[ -x, x^2] EllipticTheta[ 2, 0, x^(1/2)] EllipticTheta[ 2, Pi/4, x^(3/2)], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff(  eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^12 + A)  / (eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)), n))};

Formula

Expansion of chi(x) * psi(x) * psi(-x^3) in powers of x where psi(), chi() are Ramanujan theta functions. - Michael Somos, Nov 15 2015
Expansion of q^(-11/24) * eta(q^2)^4 * eta(q^3) * eta(q^12) / (eta(q)^2 * eta(q^4) * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 2, -2, 1, -1, 2, -2, 2, -1, 1, -2, 2, -2, ...].
a(n) = A128582(4*n + 1).
2 * a(n) = A257920(3*n + 1). - a(n) = A260118(4*n + 1). 2 * a(n) = A257921(6*n + 2). -2 * a(n) = A128580(12*n + 5) = A190615(12*n + 5). - Michael Somos, Nov 15 2015

A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 3, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2
Offset: 0

Views

Author

Michael Somos, Jul 07 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x^2 + x^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

Links

Crossrefs

Cf. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))};
  • PARI
    {a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

Formula

Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015

A261119 Expansion of f(x^2, -x^4) * f(x, x^5)^2 / f(-x^12, -x^12) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 2, 2, 0, 0, 1, 2, 4, 0, 0, 0, 0, 4, 2, 2, 0, 0, 3, 2, 2, 2, 0, 0, 2, 2, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 3, 2, 4, 2, 0, 0, 0, 6, 2, 0, 0, 0, 0, 2, 4, 0, 0, 0, 2, 2, 2, 4, 0, 0, 1, 0, 2, 0, 0, 0, 0, 2, 6, 2, 0, 0, 2, 4, 0, 2, 0, 0, 4, 4, 0, 0, 0, 0, 0, 4, 2
Offset: 0

Views

Author

Michael Somos, Aug 08 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 + 2*x^3 + x^6 + 2*x^7 + 4*x^8 + 4*x^13 + 2*x^14 + ...
G.f. = q^3 + 2*q^7 + 2*q^11 + 2*q^15 + q^27 + 2*q^31 + 4*q^35 + 4*q^55 + ...

Links

Crossrefs

Cf. A000377, A129402, A192013, A257920, A257921, A259896, A261118.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 4 n + 3}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Dec 22 2016 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A) * eta(x^24 + A) / (eta(x + A)^2 * eta(x^6 + A)^2 * eta(x^8 + A)), n))};
  • PARI
    a(n) = my(m = 4*n+3); (-1)^n*sumdiv(m, d, kronecker(12, d) * kronecker(-2, m/d)); \\ Michel Marcus, Dec 13 2017

Formula

Expansion of f(x^2, x^6) * f(x, x^5)^2 / f(x^4, x^8) in powers of x where f(,) is Ramanujan'sgeneral theta function.
Expansion of q^(-3/4) * eta(q^2)^3 * eta(q^3)^2 * eta(q^4) * eta(q^24) / (eta(q)^2 * eta(q^6)^2 * eta(q^8)) in powers of q.
Euler transform of period 24 sequence [ 2, -1, 0, -2, 2, -1, 2, -1, 0, -1, 2, -2, 2, -1, 0, -1, 2, -1, 2, -2, 0, -1, 2, -2, ...].
a(n) = (-1)^n * A257921(n) = A129402(2*n + 1) = A261118(3*n + 2) = A192013(4*n + 3) = A000377(4*n + 3).
a(2*n) = A257920(n). a(2*n + 1) = 2 * A259896(n). a(3*n) = A261118(n).
Showing 1-3 of 3 results.

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