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.

A080332 G.f.: Product_{n>0} (1 - x^n)^3 * (1 - x^(2*n - 1))^2 = Sum_{n in Z} (6*n + 1) * x^(n*(3*n + 1)/2).

Original entry on oeis.org

1, -5, 7, 0, 0, -11, 0, 13, 0, 0, 0, 0, -17, 0, 0, 19, 0, 0, 0, 0, 0, 0, -23, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 0, 0, -29, 0, 0, 0, 0, 31, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -35, 0, 0, 0, 0, 0, 37, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -41, 0, 0, 0, 0, 0, 0, 43, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -47, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Feb 18 2003

Keywords

Comments

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

Examples

			G.f. = 1 - 5*x + 7*x^2 - 11*x^5 + 13*x^7 - 17*x^12 + 19*x^15 - 23*x^22 + ...
G.f. = q - 5*q^25 + 7*q^49 - 11*q^121 + 13*q^169 - 17*q^289 + 19*q^361 + ...

References

  • J. M. Borwein, P. B. Borwein, Pi and the AGM, Wiley, 1987, p. 306.
  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.6); p. 84, Eq. (32.63).
  • S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

Links

Crossrefs

Cf. A010815, A113277.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x]^5 / QPochhammer[ x^2]^2, {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x] EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 3, 0, x], {x, 0, 2 n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[24 n + 1]}, If[ IntegerQ[ m], m KroneckerSymbol[ -3, m], 0]]; (* Michael Somos, Mar 11 2015 *)
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(24*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( (p<5) || (e%2), 0, if( p%6 == 1, p, -p)^(e\2))))}; /* Michael Somos, May 26 2005 */
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^5 / eta(x^2 + A)^2, n))};
  • PARI
    {a(n) = if( issquare( 24*n + 1, &n), n * kronecker( -3, n), 0)};

Formula

G.f.: theta_4(x)^2 * (Sum_{n in Z} (-1)^n * x^(n*(3*n + 1)/2)).
Expansion of f(-x)^2 * phi(x) = f(-x^2) * phi(-x^2)^2 in powers of x^2 where phi(), f() are Ramanujan theta functions. - Michael Somos, Feb 18 2003
Expansion of q^(-1/24) * eta(q)^5 / eta(q^2)^2 in powers of q.
Euler transform of period 2 sequence [-5, -3, ...]. - Michael Somos, Sep 09 2007
a(n) = b(24*n + 1) where b(n) is multiplicative with b(2^e) = 0^e, b(3^e) = 0^e, b(p^e) = (1+(-1)^e)/2* p^(e/2) if p == 1 (mod 6), b(p^e) = (1+(-1)^e)/2 * (-p)^(e/2) if p == 5 (mod 6). - Michael Somos, May 26 2005
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 32^(1/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A113277. - Michael Somos, Feb 18 2003
a(5*n + 3) = a(5*n + 4) = a(7*n + 3) = a(7*n + 4) = a(7*n + 6) = 0. a(25*n + 1) = -5 * a(n). - Michael Somos, Feb 18 2003

Extensions

Definition changed by N. J. A. Sloane, Aug 14 2007

A114855 Expansion of q^(-1/3) * (eta(q) * eta(q^4))^2 / eta(q^2) in powers of q.

Original entry on oeis.org

1, -2, 0, 0, 0, 4, 0, 0, -5, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, -8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 0, 0, 0, 0, 0, 0, -11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 13, 0, 0, 0, 0, 0, 0, 0, 0, -14, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 16, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, -17, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Jan 01 2006

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 4*x^5 - 5*x^8 + 7*x^16 - 8*x^21 + 10*x^33 - 11*x^40 + ...
G.f. = q - 2*q^4 + 4*q^16 - 5*q^25 + 7*q^49 - 8*q^64 + 10*q^100 - 11*q^121 + ...

References

  • S. Ramanujan, On Certain Arithmetical Functions. Collected Papers of Srinivasa Ramanujan, p. 147, Ed. G. H. Hardy et al., AMS Chelsea 2000.
  • S. Ramanujan, Notebooks, Tata Institute of Fundamental Research, Bombay 1957 Vol. 1, see page 266. MR0099904 (20 #6340)

Links

Crossrefs

Cf. A080332, A113277, A114855.

Programs

  • Magma
    A := Basis( CuspForms( Gamma1(36), 3/2), 300); A[1] - 2*A[4]; /* Michael Somos, Mar 11 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^4]^2 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Mar 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 EllipticTheta[ 2, 0, x] / (2 x^(1/4)), {x, 0, n}];  (* Michael Somos, Mar 11 2015 *)
a[ n_] := With[{m = Sqrt[3 n + 1]}, If[ IntegerQ[ m], -m (-1)^Mod[ m, 3], 0]]; (* Michael Somos, Mar 11 2015 *)
  • PARI
    {a(n) = if( issquare( 3*n + 1, &n), n * -(-1)^(n%3), 0)};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 3*n + 1; A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( e%2, 0, (-(-1)^(p%3) * p)^(e/2) )))) };
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^4 + A))^2 / eta(x^2 + A), n))};

Formula

Expansion of psi(x^2) * f(-x)^2 = phi(-x) * f(-x^4)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -2, -1, -2, -3, ...].
a(n) = b(3*n + 1) where b(n) is multiplicative and a(p^e) = 0 if e is odd, a(3^e) = 0^e, a(p^e) = p^(e/2) if p == 1 (mod 3), a(p^e) = (-p)^(e/2) if p == 2 (mod 3).
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = (u*w * (u + 2*w) * (u + 4*w))^2 - v^6 * (u^2 + 4*u*w + 8*w^2).
G.f.: Sum_{k} (3*k + 1) * x^(3*k^2 + 2*k) = Product_{k>0} (1 - x^k)^2 * (1 + x^(2*k)) * (1 - x^(4*k)).
a(4*n + 2) = a(4*n + 3) = a(8*n + 4) = 0. a(4*n + 1) = -2 * a(n). 2 * a(n) = A113277(4*n + 1) = - A114855(4*n + 1).
(-1)^n * a(n) = A113277(n). a(8*n) = A080332(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 6^(3/2) (t/i)^(3/2) f(t) where q = exp(2 Pi i t). - Michael Somos, Mar 11 2015

A230278 Expansion of q^(-2/3) * eta(q^2)^10 / eta(q)^4 in powers of q.

Original entry on oeis.org

1, 4, 4, 0, 0, -8, -16, 0, -10, -20, 16, 0, 0, 40, 0, 0, 39, 28, 0, 0, 0, -40, 32, 0, -70, 0, -64, 0, 0, -80, 0, 0, 49, -20, -40, 0, 0, 112, 80, 0, -22, 56, 64, 0, 0, 88, 0, 0, 110, -140, 0, 0, 0, 0, -160, 0, -128, 52, 0, 0, 0, -280, 0, 0, -130, 28, 156, 0, 0
Offset: 0

Views

Author

Michael Somos, Oct 15 2013

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 4*x^2 - 8*x^5 - 16*x^6 - 10*x^8 - 20*x^9 + 16*x^10 + ...
G.f. = q^2 + 4*q^5 + 4*q^8 - 8*q^17 - 16*q^20 - 10*q^26 - 20*q^29 + ...

Links

Crossrefs

Cf. A113277, A230277.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10 / QPochhammer[ q]^4, {q, 0, n}]
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / eta(x + A)^4, n))}

Formula

Expansion of psi(x^2)^2 * f(x)^4 = phi(x)^2 * f(-x^4)^4 = psi(x)^4 * f(-x^2)^2 in powers of x where phi(), psi(), f() are Ramanujan theta functions.
Euler transform of period 2 sequence [ 4, -6, ...].
G.f.: Product_{k>0} (1 - x^k)^6 * (1 + x^k)^10.
a(4*n + 3) = a(8*n + 4) = 0. 2 * a(n) = A230277(3*n + 2).
Convolution square of A113277.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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