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 1-4 of 4 results.

A101195 Expansion of psi(x^3) / psi(x) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 1, -1, 2, -3, 3, -4, 6, -7, 8, -10, 13, -16, 18, -22, 28, -33, 38, -45, 55, -65, 74, -87, 104, -121, 138, -160, 188, -217, 247, -284, 330, -378, 428, -489, 562, -640, 722, -820, 936, -1059, 1191, -1345, 1524, -1717, 1924, -2163, 2438, -2734, 3054, -3419
Offset: 0

Views

Author

Michael Somos, Dec 03 2004

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 - x^3 + 2*x^4 - 3*x^5 + 3*x^6 - 4*x^7 + 6*x^8 + ...
G.f. = q - q^5 + q^9 - q^13 + 2*q^17 - 3*q^21 + 3*q^25 - 4*q^29 + 6*q^33 + ...

Links

Crossrefs

Cf. A036018.

Programs

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

Formula

Expansion of q^(-1/2) * theta_2(q^3) / theta_2(q) in powers of q^2.
Expansion of q^(-1/4) * eta(q) * eta(q^6)^2 / (eta(q^2)^2 * eta(q^3)) in powers of q.
Given g.f. A(x), B(q) = A(q^4) * q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = u^3 - v + 3*u*v^2 - 3*u^2*v^3.
Given g.f. A(x), B(q) = (A(q^4) * q)^2 satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u^2 - v + v^2 + 3*u^2*v.
a(n) = (-1)^n * A036018(n).
G.f.: Product_{k>0} (1 + x^k + x^(2*k)) * (1 - x^k + x^(2*k))^2.
a(n) ~ (-1)^n * exp(Pi*sqrt(n/3)) / (2^(3/2) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A058483 McKay-Thompson series of class 12E for the Monster group.

Original entry on oeis.org

1, -1, 7, -9, 10, -23, 38, -47, 75, -112, 148, -217, 293, -385, 553, -728, 928, -1272, 1670, -2111, 2765, -3566, 4504, -5784, 7300, -9123, 11592, -14458, 17838, -22342, 27668, -33884, 41843, -51344, 62548, -76515, 92989, -112514, 136687, -164961, 198190, -238991
Offset: 0

Views

Author

N. J. A. Sloane, Nov 27 2000

Keywords

Comments

Given g.f. A(x), B(q) = q*A(q^2) satisfies 0 = f(B(q). B(q^2)) where f(u, v) = 12 + v^2 - 2*u^2 - u^2*v. - Michael Somos, Apr 21 2004

Examples

			G.f. = 1 - x + 7*x^2 - 9*x^3 + 10*x^4 - 23*x^5 + 38*x^6 - 47*x^7 + ...
T12E = 1/q - q + 7*q^3 - 9*q^5 + 10*q^7 - 23*q^9 + 38*q^11 - 47*q^13 + ...

Links

Crossrefs

Cf. A000521, A007240, A014708, A007241, A007267, A036018, A045478, etc.

Programs

  • Mathematica
    QP = QPochhammer; A = O[q]^50; A = ((QP[q^2 + A]^2*QP[q^3 + A])/(QP[q + A]* QP[q^6 + A]^2))^2; s = A - 3*(q/A); CoefficientList[s, q] (* Jean-François Alcover, Nov 13 2015, adapted from PARI *)
eta[q_]:= q^(1/24)*QPochhammer[q]; F:= q^(1/2)*(eta[q^2]^2*eta[q^3]/(eta[q]*eta[q^6]^2))^2; a := CoefficientList[Series[F - 3*q^1/F, {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jun 03 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x^2 * O(x^n); A = ((eta(x^2 + A)^2 * eta(x^3 + A)) / (eta(x + A) * eta(x^6 + A)^2))^2; polcoeff( A - 3*x / A, n))}; /* Michael Somos, Apr 21 2004 */

Formula

a(n) ~ (-1)^n * exp(sqrt(2*n/3)*Pi) / (2^(5/4)*3^(1/4)*n^(3/4)). - Vaclav Kotesovec, Sep 08 2017
Expansion of F - 3*q/F, where F = q^(1/2)*(eta(q^2)^2 * eta(q^3)/(eta(q) * eta(q^6)^2))^2 in powers of q. - G. C. Greubel, Jun 03 2018

A135211 Expansion of psi(-x) / psi(-x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, -1, 0, 0, -1, 0, 1, -1, 0, 2, -1, 0, 2, -2, 0, 2, -3, 0, 3, -3, 0, 4, -4, 0, 5, -6, 0, 6, -7, 0, 7, -8, 0, 10, -10, 0, 13, -13, 0, 14, -16, 0, 17, -18, 0, 22, -22, 0, 26, -28, 0, 30, -33, 0, 36, -38, 0, 44, -45, 0, 52, -55, 0, 60, -65, 0, 70, -74, 0, 84, -87, 0, 99, -104, 0, 112, -121, 0, 131, -138, 0, 156, -160
Offset: 0

Views

Author

Michael Somos, Nov 22 2007, Nov 23 2007

Keywords

Comments

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

Examples

			G.f. = 1 - x -x^4 + x^6 - x^7 + 2*x69 - x^10 + 2*x^12 - 2*x^13 + 2*x^15 + ...
G.f. = 1/q - q^3 - q^15 + q^23 - q^27 + 2*q^35 - q^39 + 2*q^47 - 2*q^51 + ...

Links

Crossrefs

Cf. A036018, A062243, A187147, A256626.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ q^(1/4) EllipticTheta[ 2, Pi/4, q^(1/2)] / EllipticTheta[ 2, Pi/4, q^(3/2)], {q, 0, n}]; (* Michael Somos, Apr 05 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) / (eta(x^2 + A) * eta(x^3 + A) * eta(x^12 + A)), n))};

Formula

Expansion of f(-x^2, -x^4) / f(x, x^5) in powers of x where f() is Ramanujan's two-variable theta function. - Michael Somos, Apr 05 2015
Expansion of q^(1/4) * eta(q) * eta(q^4) * eta(q^6) / ( eta(q^2) * eta(q^3) * eta(q^12) ) in powers of q.
Euler transform of period 12 sequence [ -1, 0, 0, -1, -1, 0, -1, -1, 0, 0, -1, 0, ...].
Given g.f. A(x), then B(q) = A(q^4) / q satisfies 0 = f(B(q), B(q^3)) where f(u, v) = (1 + v^4) - (1 + u*v)^3.
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = 3^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A036018.
a(n) = (-1)^n * A256626(n). a(3*n + 1) = - A036018(n). a(3*n + 2) = 0. - Michael Somos, Apr 05 2015
Convolution inverse is A036018. Convolution square is A062243. Convolution 4th power is A187147. - Michael Somos, Apr 05 2015

A098693 G.f.: q*Product_{k>0} (1-q^(12k))(1+q^(12k-1))(1+q^(12k-11))/(1-q^k).

Original entry on oeis.org

1, 2, 3, 5, 8, 12, 18, 26, 37, 52, 72, 99, 134, 180, 240, 317, 416, 542, 702, 904, 1158, 1476, 1872, 2364, 2973, 3724, 4647, 5778, 7160, 8844, 10890, 13370, 16368, 19984, 24336, 29561, 35822, 43308, 52242, 62884, 75536, 90552, 108342, 129384, 154232
Offset: 1

Views

Author

Ralf Stephan, Sep 21 2004

Keywords

Comments

Coefficients of a q-series of Andrews inspired by Ramanujan.

Links

Crossrefs

Cf. A036018.

Programs

  • Mathematica
    nmax = 100; Rest[CoefficientList[Series[x*Product[(1-x^(12*k)) * (1+x^(12*k-1)) * (1+x^(12*k-11))/(1-x^k), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Aug 31 2015 *)
  • PARI
    {a(n)=if(n<0, 0, polcoeff( sum(k=1, sqrtint(n), x^k^2/(1+x^k)* prod(i=1, k, (1+x^i)^2/(1-x^(2*i-1))/(1-x^(2*i)), 1+x*O(x^(n-k^2)))), n))} /* Michael Somos, Sep 19 2006 */

Formula

Euler transform of period 24 sequence [ 2, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 0, 2, 1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 0, ...]. - Michael Somos, Sep 19 2006
Given g.f. A(x), then B(x)=A(x)+A(x)^2 satisfies 0=f(B(x), B(x^2)) where f(u, v)=(1+6*u)*v*(1+2*v)-u^2. - Michael Somos, Sep 19 2006
G.f.: q*{Sum_{k} q^(24k^2+10k) +q^(24k^2+14k+1) }/{Sum_{k} (-1)^k q^((3k^2+k)/2) }. - Michael Somos, Sep 19 2006
G.f.: q*Product_{k>0} (1-q^(12k))(1+q^(12k-1))(1+q^(12k-11))/(1-q^k).
G.f.: Sum_{k>0} Prod[i=1..k, (1+q^i)^2]*(1+q^k)*q^(k^2) /{(1-q)(1-q^2)...(1-q^(2k))}.
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(9/4) * 3^(3/4) * n^(3/4)). - Vaclav Kotesovec, Aug 31 2015
Showing 1-4 of 4 results.

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