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.

A030204 Expansion of q^(-1/8) * eta(q) * eta(q^2) in powers of q.

Original entry on oeis.org

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

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Author

N. J. A. Sloane

Keywords

Comments

Number 66 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

Examples

			G.f. = 1 - x - 2*x^2 + x^3 + 2*x^5 + x^6 - 2*x^9 + x^10 - 2*x^11 - 2*x^12 + ...
G.f. = q - q^9 - 2*q^17 + q^25 + 2*q^41 + q^49 - 2*q^73 + q^81 - 2*q^89 - 2*q^97 + ...

Links

Crossrefs

Cf. A000203, A002513, A107063, A107064, A138514.

Programs

  • Magma
    Basis( CuspForms( Gamma1(128), 1), 641)[1]; /* Michael Somos, Jan 31 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x] QPochhammer[ x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 / QPochhammer[ x, x^2], {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, I x] / (4 Sqrt[ x] I^(1/4)), {x, 0, 4 n}]; (* Michael Somos, Oct 11 2013 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / (2^(3/2) x^(1/2)), {x, 0, 4 n}]; (* Michael Somos, Jan 31 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 8*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%8==1, (e + 1) * if( qfbclassno(-4*p)%8, (-1)^e, 1), e%2==0, (-1)^(e/2*(p%8<5)))))}; /* Michael Somos, Jul 26 2006 */
  • PARI
    {a(n) = if( n<0, 0, n = 8*n + 1; (qfrep([1, 0;0, 32], n) - qfrep([4, 2; 2, 9], n))[n])}; /* Michael Somos, Sep 02 2006 */

Formula

G.f.: Product_{k>0} (1 - x^k) * (1 - x^(2*k)).
G.f.: (Sum_{k>0} x^((k^2 - k)/2)) * (Sum_{k in Z} (-1)^k * x^k^2). - Michael Somos, Sep 02 2006
Expansion of psi(x) * phi(-x) = f(-x^2) * f(-x) = f(-x)^2 / chi(-x) = f(-x)^3 / phi(-x) = f(-x^2)^2 * chi(-x) = f(-x^2)^3 / psi(x) = psi(-x) * phi(-x^2) = psi(x)^2 * chi(-x)^3 = phi(-x)^2 / chi(-x)^3 = (f(-x)^3 * psi(x))^(1/2) = (f(-x^2)^3 * phi(-x))^(1/2) in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Mar 22 2008
Expansion of psi(x) * psi(-x) in powers of x^2 where psi() is a Ramanujan theta function. - Michael Somos, Oct 11 2013
Euler transform of period 2 sequence [ -1, -2, ...].
a(3*n) = A107063(n). a(3*n + 2) = -2 * A107064(n). - Michael Somos, Oct 11 2013
a(9*n + 1) = -a(n), a(9*n + 4) = a(9*n + 7) = 0. - Michael Somos, Mar 17 2004
a(n) = b(8*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = 0 if p === 3,5,7 (mod 8) and e odd, b(p^e) = (-1)^(e/2) if p == 3 (mod 8) and e even, b(p^e) = 1 if p == 5,7 (mod 8) and e even, b(p^e) = e + 1 if p == 1 (mod 8) and p = x^2 + 32*y^2, b(p^e) = (-1)^e * (e + 1) if p == 1 (mod 8) and p is not of the form x^2 + 32*y^2.
a(n) = (-1)^n * A138514(n). Convolution inverse is A002513.
G.f.: exp(Sum_{k>=1} (sigma(2*k) - 4*sigma(k))*x^k/k). - Ilya Gutkovskiy, Sep 19 2018

A083650 Expansion of f(-x, x^3) * phi(x^2) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, May 01 2003

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Essentially the expansion of eta(q)*eta(q^2). Cf. A010815. - N. J. A. Sloane, Feb 18 2010
A030204, A083650 and A138514 are the same except for signs. - N. J. A. Sloane, May 07 2010

Examples

			G.f. = 1 - x + 2*x^2 - x^3 + 2*x^5 - x^6 - 2*x^9 - x^10 + 2*x^11 - 2*x^12 - 2*x^14 + ...
G.f. = q - q^9 + 2*q^17 - q^25 + 2*q^41 - q^49 + 2*q^73 - q^81 + 2*q^89 - 2*q^97 + ...

Links

Crossrefs

Cf. A000122, A000700, A010054, A010815, A030204, A083650, A121373, A138514, A143433.

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q]*QP[q^2] + O[q]^105; Table[(-1)^Quotient[n, 2]*Coefficient[s, q, n], {n, 0, 105}] (* Jean-François Alcover, Nov 27 2015, adapted from PARI *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); (-1)^(n\2) * polcoeff( eta(x + A) * eta(x^2 + A), n))}; /* Michael Somos, Mar 02 2010 */

Formula

Euler transform of period 16 sequence [ -1, 2, 1, -2, 1, 1, -1, -3, -1, 1, 1, -2, 1, 2, -1, -2, ...].
G.f.: (Sum_{k>=0} (-1)^(k + [k/4]) * x^(k*(k+1)/2)) * (Sum_k x^(2*k^2)).
(-1)^[n/2] * a(n) = A030204(n).

Extensions

Revised by Michael Somos, Mar 02 2010

A138515 Expansion of q^(-1/4) * eta(q^2)^8 / (eta(q) * eta(q^4))^2 in powers of q.

Original entry on oeis.org

1, 2, -3, -6, 2, 0, -1, 10, 0, 2, 10, -6, -7, -14, 0, 10, -12, 0, -6, 0, 9, 4, 10, 0, 18, 2, 0, -6, -14, 18, -11, -12, 0, 0, -22, 0, 20, -14, -6, -22, 0, 0, 23, 26, 0, 18, 4, 0, -14, 2, 0, 20, 0, 0, 0, -12, 3, -30, 26, 0, -30, -14, 0, 0, 2, -30, -28, 26, 0, 18, 10, 0, -13, 34, 0, 0, 20, 0, 26, -22, 0, 6, 0, -6, 18, 0
Offset: 0

Views

Author

Michael Somos, Mar 22 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 58 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 16 2012
The weight 2 eta-quotient newform eta^8(8*z) / (eta^2(4*z)*eta^2(16*z)) appears in Theorem 2 of the Martin and Ono link in the row with conductor 64 for the strong Weil curve y^2 = x^3 - 4*x. For N(p), the number of solutions modulo primes for this elliptic curve and for y^2 = x^3 + x, see A095978. The non-vanishing p-defects p - N(p) for these two curves are given in A267859. - Wolfdieter Lang, May 26 2016

Examples

			G.f. = 1 + 2*x - 3*x^2 - 6*x^3 + 2*x^4 - x^6 + 10*x^7 + 2*x^9 + 10*x^10 - 6*x^11 + ...
G.f. for {b(n)} = q + 2*q^5 - 3*q^9 - 6*q^13 + 2*q^17 - q^25 + 10*q^29 + 2*q^37 + 10*q^41 - 6*q^45 - 7*q^49 + ...

Links

Crossrefs

Cf. A002171, A095978, A138514, A267859, A280339.

Programs

  • Magma
    A := Basis( CuspForms( Gamma0(64), 2), 342); A[1] + 2*A[3]; /* Michael Somos, May 15 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2] QPochhammer[ -q])^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)
a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^4 / (QPochhammer[ q] QPochhammer[ q^4]))^2, {q, 0, n}]; (* Michael Somos, May 15 2015 *)
  • PARI
    {a(n) = ellak( ellinit( [ 0, 0, 0, 1, 0], 1), 4*n + 1)};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^4 / (eta(x + A) * eta(x^4 + A)))^2, n))};
  • PARI
    {a(n) = my(A, p, e, x, y, a0, a1); if( n<0, 0, n = 4*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p%4==1, forstep( x=1, sqrtint(p), 2, if( issquare( p - x^2), y=x; break)); y = 2 * y * (2 - (y%4)); a0 = 1; a1 = y; for(i=2, e, x = y * a1 - p * a0; a0 = a1; a1 = x); a1, if( e%2==0, (-p)^(e / 2)))))};

Formula

Coefficients of L-series for elliptic curve "64a4": y^2 = x^3 + x.
Expansion of f(q)^2 * f(-q^2)^2 = psi(-q)^2 * phi(q)^2 = chi(q)^2 * f(-q^2)^4 = psi(q)^2 * phi(-q^2)^2 = f(q)^4 / chi(q)^2 = f(q)^6 / phi(q)^2 = f(-q^2)^6 / psi(-q)^2 = phi(q)^4 / chi(q)^6 = chi(q)^6 * psi(-q)^4 = f(q)^3 * psi(-q) = f(-q^2)^3 * phi(q) in powers of q where phi(), psi(), chi(), f() are Ramanujan theta functions.
Euler transform of period 4 sequence [2, -6, 2, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (64 t)) = 64 (t/i)^2 f(t) where q = exp(2 Pi i t).
a(n) = b(4*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(p^e) = (1 + (-1)^e)/2 * (-p)^(e/2) if p == 3 (mod 4), b(p^e) = b(p) * b(p^(e-1)) - p * b(p^(e-2)) if p == 1 (mod 4) with b(p) = 2 * x * (-1)^((x-1)/2) where p = x^2 + 4 * y^2.
G.f.: (Product_{k>0} (1 - x^(2*k))^2 * (1 + x^(2*k - 1)))^2.
a(n) = (-1)^n * A002171(n). a(9*n + 2) = -3 * a(n), a(9*n + 5) = a(9*n + 8) = 0. Convolution square of A138514.
G.f. for{b(n)}:
eta^8(8*z)/(eta^2(4*z)*eta^2(16*z)) with q = exp(2*Pi*i*z)), Im(z) > 0 (see a comment on the Martin-Ono link above). - Wolfdieter Lang, May 27 2016
Showing 1-3 of 3 results.

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

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