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 1-7 of 7 results.

A375107 Expansion of Sum_{k in Z} x^(2*k) / (1 - x^(7*k+3)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 30 2024

Keywords

Links

Crossrefs

Cf. A375106, A375108.

Programs

  • PARI
    my(N=100, x='x+O('x^N)); Vec(sum(k=-N, N, x^(2*k)/(1-x^(7*k+3))))
  • PARI
    my(N=100, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2/((1-x^(7*k-3))*(1-x^(7*k-4)))))

Formula

G.f.: Product_{k>0} (1-x^(7*k))^2 / ((1-x^(7*k-3)) * (1-x^(7*k-4))).
G.f.: Sum_{k in Z} x^(3*k) / (1 - x^(7*k+2)).

A375148 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+2)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 01 2024

Keywords

Crossrefs

Cf. A374900, A375106, A375149, A375150.
Cf. A033687, A340453.

Programs

  • PARI
    my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+2))))
  • PARI
    my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-3))*(1-x^(7*k-4)))^2/(1-x^k)))

Formula

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2 / (1-x^k).
G.f.: Sum_{k in Z} x^(2*k) / (1 - x^(7*k+1)).

A375149 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+4)).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 1, 1, 0, 1, 1, 0, 3, 1, -1, 1, 2, 0, 0, 1, 1, 1, 1, 1, 2, 1, -1, 1, 2, 0, 1, 0, 0, 1, 2, 0, 3, 1, -1, 1, 2, 0, 1, 1, 0, 2, 1, 0, 0, 1, -1, 1, 3, 1, 1, 1, 2, 0, 0, 0, 3, 1, -1, 1, 2, -1, 1, 2, 0, 1, 0, 0, 2, 1, 0, 1, 3, 0, 2, 1, 0, 1, 0, 0, 3, 1, -1, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 1, -1, 0, 5
Offset: 0

Views

Author

Seiichi Manyama, Aug 01 2024

Keywords

Crossrefs

Cf. A374900, A375106, A375148, A375150.

Programs

  • PARI
    my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+4))))
  • PARI
    my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^3*((1-x^(7*k-2))*(1-x^(7*k-5)))^2/(1-x^k)))

Formula

G.f.: Product_{k>0} (1-x^(7*k))^3 * ((1-x^(7*k-2)) * (1-x^(7*k-5)))^2 / (1-x^k).
G.f.: Sum_{k in Z} x^(4*k) / (1 - x^(7*k+1)).

A375150 Expansion of Sum_{k in Z} x^k / (1 - x^(7*k+5)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Aug 01 2024

Keywords

Crossrefs

Cf. A374900, A375106, A375148, A375149.
Cf. A375107.

Programs

  • PARI
    my(N=110, x='x+O('x^N)); Vec(sum(k=-N, N, x^k/(1-x^(7*k+5))))
  • PARI
    my(N=110, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2/((1-x^(7*k-2))*(1-x^(7*k-5)))))

Formula

G.f.: Product_{k>0} (1-x^(7*k))^2 / ((1-x^(7*k-2)) * (1-x^(7*k-5))).
G.f.: Sum_{k in Z} x^(5*k) / (1 - x^(7*k+1)).

A108483 Expansion of f(-x^2, -x^5) / f(-x, -x^6) in powers of x where f() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 0, 0, 0, -1, 0, 1, 1, 0, -1, -1, -1, 0, 2, 2, 0, -1, -2, -2, 0, 3, 3, 0, -2, -3, -3, 0, 5, 5, 1, -3, -5, -5, 0, 7, 7, 1, -5, -8, -7, 1, 11, 12, 2, -7, -12, -11, 1, 15, 16, 3, -11, -18, -15, 2, 23, 24, 5, -15, -26, -22, 3, 31, 33, 7, -22, -37, -30, 5, 44, 47, 11, -30, -52, -42, 6, 59, 63, 15, -42, -72, -56, 10, 82, 88, 22
Offset: 0

Views

Author

Michael Somos, Jun 04 2005

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
In Duke (2005) page 157 the g.f. is denoted by t(tau).

Examples

			G.f. = 1 + x - x^5 + x^7 + x^8 - x^10 - x^11 - x^12 + 2*x^14 + 2*x^15 + ...
G.f. = q^-2 + q^5 - q^33 + q^47 + q^54 - q^68 - q^75 - q^82 + 2*q^96 + ...

References

  • W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162.

Links

Crossrefs

Cf. A229894, A375106, A375150.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^2, x^7] QPochhammer[ x^5, x^7] / (QPochhammer[ x, x^7] QPochhammer[ x^6, x^7]), {x, 0, n}]; (* Michael Somos, Oct 03 2013 *)
a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{-1, 1, 0, 0, 1, -1, 0}[[Mod[k, 7, 1]]], {k, n}], {x, 0, n}]; (* Michael Somos, May 03 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k + x*O(x^n))^[ 0, -1, 1, 0, 0, 1, -1][k%7 + 1]), n))};

Formula

Euler transform of period 7 sequence [ 1, -1, 0, 0, -1, 1, 0, ...]. - Michael Somos, Oct 03 2013
Given g.f. A(x), then B(q) = q^-2*A(q^7) satisfies 0 = f(B(q), B(q^2)) where f(u, v) = v^3 - u^6 + 3*u^4*v + u^7*v^3 + u^2*v^9 + u^8*v^6 - 3*u^2*v^2 - 2*u*v^6 - 5*u^3*v^5 - u^5*v^4 - u^9*v^2 - u^4*v^8 - u^6*v^7.
G.f.: Product_{k>0} (1 - x^(7*k - 2)) * (1 - x^(7*k - 5)) / ((1 - x^(7*k - 1)) * (1 - x^(7*k - 6))).
a(n) = A229894(7*n). - Michael Somos, Oct 03 2013
G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375150. - Seiichi Manyama, Aug 03 2024

A375108 Expansion of Sum_{k in Z} x^(3*k) / (1 - x^(7*k+3)).

Original entry on oeis.org

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

Views

Author

Seiichi Manyama, Jul 30 2024

Keywords

Links

Crossrefs

Cf. A375106, A375107.

Programs

  • PARI
    my(N=100, x='x+O('x^N)); Vec(sum(k=-N, N, x^(3*k)/(1-x^(7*k+3))))
  • PARI
    my(N=100, x='x+O('x^N)); Vec(prod(k=1, N, (1-x^(7*k))^2*(1-x^(7*k-1))*(1-x^(7*k-6))/((1-x^(7*k-3))*(1-x^(7*k-4)))^2))

Formula

G.f.: Product_{k>0} (1-x^(7*k))^2 * (1-x^(7*k-1)) * (1-x^(7*k-6)) / ((1-x^(7*k-3)) * (1-x^(7*k-4)))^2.

A108482 Expansion of a modular function for Gamma(7).

Original entry on oeis.org

1, 1, 1, 0, -1, -1, 0, 1, 2, 1, -1, -3, -3, 0, 3, 5, 3, -2, -6, -6, -1, 6, 9, 5, -4, -12, -11, -1, 12, 18, 10, -7, -21, -21, -3, 20, 30, 17, -13, -37, -35, -4, 36, 53, 30, -20, -62, -59, -8, 57, 85, 47, -35, -101, -95, -11, 94, 138, 78, -54, -159, -150, -19, 145, 213, 118, -85, -247, -231, -27, 225, 330, 183, -128, -375
Offset: 0

Views

Author

Michael Somos, Jun 04 2005

Keywords

References

  • W. Duke, Continued fractions and modular functions, Bull. Amer. Math. Soc. 42 (2005), 137-162. See page 157.

Crossrefs

Cf. A375106, A375107.

Programs

  • PARI
    {a(n)=if(n<0, 0, polcoeff( prod(k=1,n,(1-x^k+x*O(x^n))^[0,-1,0,1,1,0,-1][k%7+1]), n))}

Formula

Given g.f. A(x), then B(x)=x^-3*A(x^7) satisfies 0=f(B(x), B(x^2)) where f(u, v)=u^7 -v^7 +u*v^3 +u^9*v^6 +u^2*v^6 +3*u^5*v^8 -u^3*v^2 -u^3*v^9 -u^4*v^5 -u^5*v -5*u^6*v^4 -3*u^7*v^7 -2*u^8*v^3.
G.f.: Product_{k>0} (1-x^(7k-3))(1-x^(7k-4))/((1-x^(7k-1))(1-x^(7k-6))).
G.f.: B(x) / C(x), where B(x) is the g.f. of A375106 and C(x) is the g.f. of A375107. - Seiichi Manyama, Aug 03 2024
Showing 1-7 of 7 results.

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