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.

A192323 Expansion of theta_3(q^3) * theta_3(q^5) in powers of q.

Original entry on oeis.org

1, 0, 0, 2, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 4, 0, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 6, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 4, 0, 0, 2, 0, 0, 0, 0, 6, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 4, 4, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Aug 01 2011

Keywords

Examples

			G.f. = 1 + 2*q^3 + 2*q^5 + 4*q^8 + 2*q^12 + 4*q^17 + 2*q^20 + 4*q^23 + 2*q^27 + ...

Links

Crossrefs

Cf. A028956, A122855, A140727, A260671.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] EllipticTheta[ 3, 0, q^5], {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, qfrep([3, 0; 0, 5], n)[n]*2)};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^5 / (eta(x^3 + A) * eta(x^5 + A) * eta(x^12 + A) * eta(x^20 + A))^2, n))};

Formula

Expansion of (eta(q^6) * eta(q^10))^5 / (eta(q^3) * eta(q^5) * eta(q^12) * eta(q^20))^2 in powers of q.
Euler transform of a period 60 sequence.
G.f.: (Sum_{k} x^(3 * k^2)) * (Sum_{k} x^(5 * k^2)).
a(3*n + 1) = a(4*n + 2) = a(5*n + 1) = a(5*n + 4) = 0. a(4*n) = A028956(n).
a(n) = A122855(n) - A140727(n). a(5*n) = A260671(n).

A260671 Expansion of theta_3(q) * theta_3(q^15) in powers of q.

Original entry on oeis.org

1, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 6, 0, 0, 4, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 2, 4, 0, 0, 10, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 4, 0
Offset: 0

Views

Author

Michael Somos, Nov 14 2015

Keywords

Comments

a(n) is the number of solutions in integers (x, y) of x^2 + 15*y^2 = n. - Michael Somos, Jul 17 2018

Examples

			G.f. = 1 + 2*x + 2*x^4 + 2*x^9 + 2*x^15 + 6*x^16 + 4*x^19 + 4*x^24 + 2*x^25 + ...

Links

Crossrefs

Cf. A028625, A122855, A140727, A192323, A260675, A260676.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^15], {q, 0, n}];
  • PARI
    {a(n) = if( n<1, n==0, qfrep([1, 0; 0, 15], n)[n]*2)};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^5 / (eta(x + A) * eta(x^4 + A) * eta(x^15 + A) * eta(x^60 + A))^2, n))};
  • PARI
    q='q+O('q^99); Vec((eta(q^2)*eta(q^30))^5/(eta(q)*eta(q^4)*eta(q^15)*eta(q^60))^2) \\ Altug Alkan, Jul 18 2018

Formula

Expansion of (eta(q^2) * eta(q^30))^5 / (eta(q) * eta(q^4) * eta(q^15) * eta(q^60))^2 in powers of q.
Euler transform of a period 60 sequence.
G.f. is a period 1 Fourier series which satisfies f(-1 / (60 t)) = 60^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: (Sum_{k in Z} x^(k^2)) * (Sum_{k in Z} x^(15*k^2)).
a(3*n + 2) = a(4*n + 2) = a(5*n + 2) = a(5*n + 3) = 0.
a(4*n) = A028625(n). a(4*n + 1) = 2 * A260675(n). a(4*n + 3) = 2 * A260676(n).
a(5*n) = A192323(n).
a(n) = A122855(n) + A140727(n).

A140728 Expansion of (phi(-q^3) * phi(-q^5) - phi(-q) * phi(-q^15)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, -1, -1, -1, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, -3, -2, 0, 2, 1, 0, 0, -2, -2, 1, 0, -1, 0, 0, 0, 2, 4, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, -1, 0, -2, 3, 1, 0, 2, 0, -2, 0, 0, 0, -2, 0, 0, -1, 2, 0, 0, -5, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, -1, -2, 0, 0, 2, 3, 1, 0, -2, 0, 2, 0, 0, 0, 0, 0, 0, 2, -2, 0, -2, -4, 0, 0, 0, -1, 0, 0, 0, 0, 0
Offset: 1

Views

Author

Michael Somos, May 29 2008

Keywords

Comments

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

Examples

			G.f. = q - q^3 - q^4 - q^5 + 2*q^8 + q^9 + q^12 + q^15 - 3*q^16 - 2*q^17 + ...

Links

Crossrefs

Cf. A122855, A140727.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, -(-1)^# KroneckerSymbol[ 5, #] KroneckerSymbol[ -3, n/#] &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ q QPochhammer[ q^2] QPochhammer[ q^30] QPochhammer[ q^3, q^6] QPochhammer[ q^5, q^10], {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, q^3] EllipticTheta[ 4, 0, q^5] - EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^15]) / 2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, -(-1)^d * kronecker(5, d) * kronecker(-3, n/d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, (-1)^e * (1-e), p==3 || p==5, (-1)^e, kronecker(p, 15)==1, (e+1) * (-1)^(e*valuation(p%15, 2)), (1 + (-1)^e) / 2)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A) * eta(x^5 + A) * eta(x^30 + A) / (eta(x^6 + A) * eta(x^10 + A)), n))};

Formula

Expansion of q * f(-q^2) * f(-q^30) * chi(-q^3) * chi(-q^5) in powers of q where f(), chi() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3) * eta(q^5) * eta(q^30) / (eta(q^6) * eta(q^10)) in powers of q.
Euler transform of period 30 sequence [0, -1, -1, -1, -1, -1, 0, -1, -1, -1, 0, -1, 0, -1, -2, -1, 0, -1, 0, -1, -1, -1, 0, -1, -1, -1, -1, -1, 0, -2, ...].
a(n) is multiplicative with a(2^e) = (-1)^e * (1-e) if e > 0. a(3^e) = a(5^e) = (-1)^e, a(p^e) = e+1 if p == 1, 4 (mod 15), a(p^e) = (-1)^e * (e+1) if p == 2, 8 (mod 15), a(p^e) = (1 + (-1)^e) / 2 if p == 7, 11, 13, 14 (mod 15).
G.f. is a period 1 Fourier series which satisfies f(-1 / (30 t)) = 60^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A121362.
G.f.: x * Product_{k>0} (1 - x^(2*k)) * (1 - x^(30*k)) / ((1 + x^(3*k)) * (1 + x^(5*k))).
G.f.: Sum_{k>0} Kronecker(5, n) * x^n / (1 - x^n + x^(2*n)) = Sum_{k>0} -(-1)^n * Kronecker(5, n) * x^n / (1 + x^n + x^(2*n)).
a(n) = -(-1)^n * A140727(n). abs(a(n)) = A122855(n).

A281640 Expansion of x * f(x, x) * f(x^5, x^25) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 2, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 3, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 6, 0, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 1, 4, 0, 0, 2, 0, 2
Offset: 1

Views

Author

Michael Somos, Jan 25 2017

Keywords

Examples

			G.f. = x + 2*x^2 + 2*x^5 + x^6 + 2*x^7 + 4*x^10 + 2*x^15 + 2*x^17 + 2*x^22 + ...
G.f. = q^5 + 2*q^8 + 2*q^17 + q^20 + 2*q^23 + 4*q^32 + 2*q^47 + 2*q^53 + 2*q^65 + ...

Links

Crossrefs

Cf. A122855, A122856, A140727, A192323, A258276, A260649.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ 3 n + 2, KroneckerSymbol[ -15, #] (-1)^Boole[Mod[#, 4] == 2] &]];
a[ n_] := SeriesCoefficient[ x EllipticTheta[ 3, 0, x] QPochhammer[ -x^5, x^30] QPochhammer[ -x^25, x^30] QPochhammer[ x^30], {x, 0, n}];
  • PARI
    {a(n) = if( n<1, 0, sumdiv(3*n + 2, d, kronecker(-15, d) * (-1)^(d%4==2) ))};
  • PARI
    {a(n) = if( n<1, 0, my(m = 3*n + 2, s, x); for(y=1, sqrtint(m\5), if( y%3 && issquare((m - 5*y^2)\3, &x), s += (x>0) + 1)); s)};
  • PARI
    {a(n) = if( n<1, 0, my(A); n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^10 + A)^2 * eta(x^15 + A) * eta(x^60 + A) / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^5 + A) * eta(x^20 + A) * eta(x^30 + A)), n))};

Formula

G.f.: x * (Sum_{k in Z} x^k^2) * (Sum_{k in Z} x^(15*k^2 - 10*k)).
G.f.: x * Product_{k>0} (1 + x^(2*k-1))^2 * (1 - x^(2*k)) * (1 + x^(30*k-25)) * (1 + x^(30*k-5)) * (1 - x^(30*k)).
a(n) = A122855(3*n + 2) = A260649(3*n + 2) = A122856(6*n + 4) = A258276(6*n + 4).
a(n) = - A140727(3*n + 2). 2 * a(n) = A192323(3*n + 2).
Showing 1-4 of 4 results.

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