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.

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A258832 Expansion of psi(-x^3) * f(-x, x^2) in powers of x where psi(), f(,) are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jun 11 2015

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 + x^4 - 2*x^5 - x^7 + x^8 - x^9 + 2*x^10 + ...
G.f. = q^5 - q^17 + q^29 - q^41 + q^53 - 2*q^65 - q^89 + q^101 - q^113 + ...

Links

Crossrefs

Cf. A121444, A258831.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 12 n + 5, KroneckerSymbol[ -4, #] &] / 2];
a[ n_] := SeriesCoefficient[ QPochhammer[ x^6]^2 QPochhammer[ x, -x] / QPochhammer[ x^3, -x^3], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv( 12*n + 5, d, kronecker( -4, d)) / 2)};
  • 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)^4 / (eta(x^2 + A)^2 * eta(x^3 + A) * eta(x^12 + A)), n))};

Formula

Expansion of q^(-5/12) * eta(q) * eta(q^4) * eta(q^6)^4 / (eta(q^2)^2 * eta(q^3) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ -1, 1, 0, 0, -1, -2, -1, 0, 0, 1, -1, -2, ...].
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(3*k))^2 * (1 - x^k + x^(2*k))^3 / (1 - x^(2*k) + x^(4*k)).
a(n) = (-1)^n * A121444(n). Convolution square is A258831.

A256014 Expansion of phi(-q^3)^4 / (phi(-q) * phi(-q^9)) in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 4, 0, -2, -8, 0, 0, 4, -4, -4, 0, 0, 4, 0, 0, -2, -8, 4, 0, 8, 0, 0, 0, 0, 6, 8, 0, 0, -8, 0, 0, 4, 0, -4, 0, 4, 4, 0, 0, -4, -8, 0, 0, 0, -8, 0, 0, 0, 2, 12, 0, -4, -8, 0, 0, 0, 0, -4, 0, 0, 4, 0, 0, -2, -16, 0, 0, 8, 0, 0, 0, 4, 4, 8, 0, 0, 0, 0, 0, 8
Offset: 0

Views

Author

Michael Somos, Jun 03 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 4*q^2 - 2*q^4 - 8*q^5 + 4*q^8 - 4*q^9 - 4*q^10 + 4*q^13 + ...

Links

Crossrefs

Cf. A002175, A104794, A121444, A122856, A122865, A256280, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q^3]^4 / (EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^9]), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^8 * eta(x^18 + A) / (eta(x + A)^2 * eta(x^6 + A)^4 * eta(x^9 + A)^2), n))};
  • PARI
    {a(n) = if( n<1, n==0, 2^(n%3) * (-1)^(n\3) * sumdiv(n, d, [0, 1, 2, -1][d%4 + 1] * if(d%9, 1, 4) * (-1)^((d%8==6) + n+d)))};

Formula

Expansion of eta(q^2) * eta(q^3)^8 * eta(q^18) / (eta(q)^2 * eta(q^6)^4 * eta(q^9)^2) in powers of q.
Euler transform of period 18 sequence [ 2, 1, -6, 1, 2, -3, 2, 1, -4, 1, 2, -3, 2, 1, -6, 1, 2, -2, ...].
a(n) = (-1)^n * A256280(n). a(3*n + 1) = 2 * A258277(n). a(3*n + 2) = 4 * A258278(n). a(4*n) = A256280(n). a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0.
a(6*n + 2) = 4 * A122865(n). a(6*n + 4) = -2 * A122856(n). a(9*n) = A104794(n). a(12*n + 1) = A002175(n). a(12*n + 5) = -8 * A121444(n).

A259660 Expansion of f(-x, -x^11) * psi(-x^3)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 02 2015

Keywords

Comments

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

Examples

			G.f. = 1 - x^3 + x^4 + x^5 + x^8 - x^11 + x^12 + x^15 + x^16 - x^17 - x^19 + ...
G.f. = q^5 - q^14 + q^17 + q^20 + q^29 - q^38 + q^41 + q^50 + q^53 - q^56 + ...

Links

Crossrefs

Cf. A121444, A247133, A259529.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ Product[ (1 - x^k)^{ 0, 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0}[[Mod[ k, 12, 1]]], {k, n}], {x, 0, n}];
QP:= QPochhammer; a[n_]:= SeriesCoefficient[(QP[x, x^12]*QP[x^11,x^12]* QP[x^12]*QP[x^3, -x^3]^2*QP[x^6]^2)/(QP[x, -x]*QP[x^2]), {x, 0, n}]; Table[a[n], {n, 0, 100}] (* G. C. Greubel, Mar 17 2018 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( prod(k=1, n, (1 - x^k)^([ 2, 0, 0, 1, -1, -1, 0, -1, -1, 1, 0, 0][k%12 + 1]), 1 + x * O(x^n)), n))};

Formula

Expansion of f(-x, -x^11) * f(x, x^5)^2 / f(x) in powers of x where f(,) is the Ramanujan general theta function.
Euler transform of period 12 sequence [ 0, 0, -1, 1, 1, 0, 1, 1, -1, 0, 0, -2, ...].
a(4*n) = A121444(n). a(4*n + 1) = a(n - 1). a(4*n + 2) = 0.
Convolution of A247133 and A259529.

A329958 Expansion of q^(-13/24) * eta(q^2)^3 * eta(q^3) * eta(q^6) / eta(q)^2 in powers of q.

Original entry on oeis.org

1, 2, 2, 3, 3, 4, 4, 3, 5, 3, 6, 7, 4, 5, 4, 8, 6, 5, 7, 6, 7, 8, 7, 5, 8, 10, 9, 4, 7, 7, 9, 11, 8, 10, 5, 10, 12, 7, 10, 8, 10, 12, 4, 10, 8, 13, 15, 10, 9, 5, 15, 9, 12, 11, 10, 12, 10, 11, 11, 12, 15, 12, 6, 14, 8, 11, 17, 13, 12, 9, 16, 17, 8, 15, 10, 14
Offset: 0

Views

Author

Michael Somos, Nov 26 2019

Keywords

Examples

			G.f. = 1 + 2*x + 2*x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 4*x^6 + 3*x^7 + 5*x^8 + ...
G.f. = q^13 + 2*q^37 + 2*q^61 + 3*q^85 + 3*q^109 + 4*q^133 + 4*q^157 + ...

Crossrefs

Cf. A010054, A033762, A080995, A121444, A329955.

Programs

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

Formula

Euler transform of period 6 sequence [2, -1, 1, -1, 2, -3, ...].
G.f.: Product_{k>=1} (1 + x^k)^2 * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 - x^(6*k)).
Convolution of A033762 and A080995. Convolution of A010054 and A121444.
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = (3/2)^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A329955.
Previous Showing 21-24 of 24 results.

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