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.

A093950 Expansion of 1 / (chi(-x) * chi(-x^7)) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 2, 2, 3, 4, 6, 7, 9, 12, 14, 18, 22, 28, 34, 41, 50, 60, 72, 86, 105, 124, 146, 174, 204, 240, 282, 332, 386, 450, 524, 606, 703, 812, 940, 1082, 1243, 1428, 1636, 1873, 2140, 2448, 2788, 3172, 3610, 4096, 4646, 5264, 5962, 6736, 7606, 8582, 9666, 10884
Offset: 0

Views

Author

Michael Somos, Apr 19 2004

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the right side of Cayley's identity is 2 * q * A(q^2). - Michael Somos, Dec 03 2013
Proof of Cayley's identity, from Silviu Radu, Mar 13 2015: (Start)
Up to issues of convergence I observe that the identity may be rewritten after substituting q=e^{2 Pi Iz} as:
E(28z)^(-1) x E(14z)^2 x E(7z)^(-1) x E(4z)^(-1) x E(2z)^2 x E(z)^(-1) -E(14z)^(-1) x E(7z) x E(2z)^(-1) x E(z) = 2 E(28z) x E(14z)^(-1) x E(4z) x E(2z)^(-1)
where E(z)= exp( Pi I z/12) Product_{n>=1} (1-e^{2 Pi I z n}) is the Dedekind eta function.
One can further rewrite the above identity by dividing the whole identity by the first term. We obtain:
1-E(28z) x E(14z)^(-3) x E(7z)^2 x E(4z) x E(2z)^(-3) x E(z)^2
-2 E(28z)^2 x E(14z)^(-3) x E(7z) x E(4z)^2 x E(2z)^(-3) x E(z)=0
What is interesting now about this expression is that each term is a modular function for the group Gamma_0(28).
Furthermore, all the terms except the constant term have two poles, therefore the whole left hand side has at most two poles (at the points z=1/14 and z=1/2).
However we check that in the q-expansion the first three coefficients are zero, which implies that the left hand side also has a zero of order at least three at the point infinity (note that z=I x infty transforms into q=0, q=e^(2 Pi iz} ).
It is impossible that a nonzero modular function has more zeros than poles, therefore it is the zero function. This finishes the proof. (End)

Examples

			G.f. = 1 + x + x^2 + 2*x^3 + 2*x^4 + 3*x^5 + 4*x^6 + 6*x^7 + 7*x^8 + ...
G.f. = q + q^4 + q^7 + 2*q^10 + 2*q^13 + 3*q^16 + 4*q^19 + 6*q^22 + ...

References

  • A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

Crossrefs

Cf. A102314, A112212, A246762.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^7, x^7], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + x^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + x^(7*k), 1 + x * O(x^n)), n))};
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^14 + A) / (eta(x + A) * eta(x^7 + A)), n))};

Formula

Expansion of q^(-1/3) * (eta(q^2) * eta(q^14)) / (eta(q) * eta(q^7)) in powers of q.
Euler transform of period 14 sequence [ 1, 0, 1, 0, 1, 0, 2, 0, 1, 0, 1, 0, 1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v - 2*u*v^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 1/2 * g(t) where q = exp(2 Pi i t) and g() is the g.f. of A102314. - Michael Somos, Dec 03 2013
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(7*k)).
a(n) = A112212(2*n + 1) = - A102314(2*n + 1). - Michael Somos, Dec 03 2013
Convolution inverse of A102314.
a(n) = (-1)^n * A246762(n). - Michael Somos, Sep 02 2014
a(n) ~ exp(2*Pi*sqrt(2*n/21)) / (2^(7/4) * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

Extensions

Entry revised by N. J. A. Sloane, Mar 15 2015 (with thanks to Doron Zeilberger)

A102314 McKay-Thompson series of class 42C for the Monster group.

Original entry on oeis.org

1, -1, 0, -1, 1, -1, 1, -2, 3, -2, 3, -3, 4, -4, 4, -6, 7, -7, 7, -9, 10, -12, 13, -14, 17, -18, 19, -22, 26, -28, 29, -34, 38, -41, 44, -50, 57, -60, 65, -72, 81, -86, 94, -105, 114, -124, 133, -146, 161, -174, 187, -204, 224, -240, 258, -282, 309, -332, 354, -386, 419, -450, 481, -524, 569, -606, 651, -703
Offset: 0

Views

Author

Michael Somos, Jan 03 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Given g.f. A(x), the second term of the left side of Cayley's identity is -A(q). - Michael Somos, Dec 03 2013

Examples

			G.f. = 1 - x - x^3 + x^4 - x^5 + x^6 - 2*x^7 + 3*x^8 - 2*x^9 + 3*x^10 - 3*x^11 + ...
T42C = 1/q - q^2 - q^8 + q^11 - q^14 + q^17 - 2*q^20 + 3*q^23 - 2*q^26 + ...

References

  • A. Cayley, An elliptic-transcendant identity, Messenger of Math., 2 (1873), p. 179.

Links

Crossrefs

Cf. A093950, A112212, A193826, A193883.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x, x^2] QPochhammer[ x^7, x^14], {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
a[ n_] := SeriesCoefficient[ 1 / ( Product[ 1 + x^k, {k, n}] Product[ 1 + x^k, {k, 7, n, 7}] ), {x, 0, n}]; (* Michael Somos, Aug 06 2011 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^7 + A) / (eta(x^2 + A) * eta(x^14 + A)), n))};

Formula

Expansion of chi(-x) * chi(-x^7) in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/3) * eta(q) * eta(q^7) / (eta(q^2) * eta(q^14)) in powers of q.
Euler transform of period 14 sequence [ -1, 0, -1, 0, -1, 0, -2, 0, -1, 0, -1, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^2 - u^2*v - 2*u.
G.f. is a period 1 Fourier series which satisfies f(-1 / (126 t)) = 2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A093950.
G.f.: 1 / (Product_{k>0} (1 + x^k) * (1 + x^(7*k))).
a(n) = (-1)^n * A112212(n). a(2*n + 1) = - A093950(n). a(4*n) = A193826(n). a(4*n + 2) = A193883(n).
Convolution inverse is A093950.
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/21)) / (2 * 21^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2017

A246762 Expansion of 1 / (chi(x) * chi(x^7)) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -2, 2, -3, 4, -6, 7, -9, 12, -14, 18, -22, 28, -34, 41, -50, 60, -72, 86, -105, 124, -146, 174, -204, 240, -282, 332, -386, 450, -524, 606, -703, 812, -940, 1082, -1243, 1428, -1636, 1873, -2140, 2448, -2788, 3172, -3610, 4096, -4646, 5264, -5962
Offset: 0

Views

Author

Michael Somos, Sep 02 2014

Keywords

Comments

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

Examples

			G.f. = 1 - x + x^2 - 2*x^3 + 2*x^4 - 3*x^5 + 4*x^6 - 6*x^7 + 7*x^8 - 9*x^9 + ...
G.f. = q - q^4 + q^7 - 2*q^10 + 2*q^13 - 3*q^16 + 4*q^19 - 6*q^22 + 7*q^25 + ...

Links

Crossrefs

Cf. A093950, A112212.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ 1 + (-x)^k, {k, n}] Product[ 1 + (-x)^k, {k, 7, n, 7}], {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x] QPochhammer[ x^7, -x^7], {x, 0, n}];
eta[q_]:= q^(1/24)*QPochhammer[q]; a:= CoefficientList[Series[q^(-1/3)* eta[q]*eta[q^4]*eta[q^7]*eta[q^28]/(eta[q^2]*eta[q^14])^2, {q,0,60}],q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jul 04 2018 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod( k=1, n, 1 + (-x)^k, 1 + x * O(x^n)) * prod( k=1, n\7, 1 + (-x)^(7*k), 1 + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^7 + A) * eta(x^28 + A) / (eta(x^2 + A) * eta(x^14 + A))^2, n))};

Formula

Expansion of q^(-1/3) * eta(q) * eta(q^4) * eta(q^7) * eta(q^28) / (eta(q^2) * eta(q^14))^2 in powers of q.
Euler transform of period 28 sequence [ -1, 1, -1, 0, -1, 1, -2, 0, -1, 1, -1, 0, -1, 2, -1, 0, -1, 1, -1, 0, -2, 1, -1, 0, -1, 1, -1, 0, ...].
Given g.f. A(x), then B(q) = q * A(q^3) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = (u - v^2) * (v - u^2) - 2 * (u*v)^2 * (1 - u*v)^2.
G.f. is a period 1 Fourier series which satisfies f(-1 / (252 t)) = f(t) where q = exp(2 Pi i t).
G.f.: Product_{k>0} (1 + (-x)^k) * (1 + (-x)^(7*k)).
a(n) = (-1)^n * A093950(n).
Convolution inverse of A112212.
Showing 1-3 of 3 results.

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