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.

A128617 Expansion of q^2 * psi(q) * psi(q^15) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Mar 13 2007

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Also the number of positive odd solutions to equation x^2 + 15y^2 = 8n. - Seiichi Manyama, May 21 2017

Examples

			G.f. = x^2 + x^3 + x^5 + x^8 + x^12 + 2*x^17 + x^18 + x^20 + 2*x^23 + x^27 + x^30 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(i).

Links

Crossrefs

Cf. A035162.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] - KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q^2 (QPochhammer[ q^2] QPochhammer[ q^30])^2 / (QPochhammer[ q] QPochhammer[ q^15]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) - kronecker(20, d) * kronecker(-3, n/d) )/2)};
  • PARI
    {a(n) = my(A); if( n<2, 0, n-=2; A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^30 + A))^2 / (eta(x + A) * eta(x^15 + A)), n))};

Formula

Expansion of (eta(q^2) * eta(q^30))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 30 sequence [ 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 2, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -1, 1, -2, ...].
For n>0, n in A028955 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = -A121362(n).
a(n)= (A082451(n) - A121362(n) )/2.
G.f.: x^2 * Product_{k>0} (1 - x^k) * (1 - x^(15*k)) * (1 + x^(2*k))^2 * (1 + x^(30*k))^2.

A128616 Expansion of q * psi(q^3) * psi(q^5) in powers of q where psi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Mar 13 2007

Keywords

Comments

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

Examples

			G.f. = x + x^4 + x^6 + x^9 + x^10 + x^15 + x^16 + 2*x^19 + x^24 + x^25 + 2*x^31 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 377, Entry 9(iv).

Links

Crossrefs

Cf. A082451, A121362.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -60, #] + KroneckerSymbol[ 20, #] KroneckerSymbol[ -3, n/#] &] / 2]; (* Michael Somos, Nov 12 2015 *)
a[ n_] := SeriesCoefficient[ q(QPochhammer[ q^6] QPochhammer[ q^10])^2 / (QPochhammer[ q^3] QPochhammer[ q^5]), {q, 0, n}]; (* Michael Somos, Nov 12 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, kronecker(-60, d) + kronecker(20, d) * kronecker(-3, n/d) )/2)};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x^6 + A) * eta(x^10 + A))^2 / (eta(x^3 + A) * eta(x^5 + A)), n))};

Formula

Expansion of (eta(q^6) * eta(q^10))^2 / (eta(q^3) * eta(q^5)) in powers of q.
Euler transform of period 30 sequence [ 0, 0, 1, 0, 1, -1, 0, 0, 1, -1, 0, -1, 0, 0, 2, 0, 0, -1, 0, -1, 1, 0, 0, -1, 1, 0, 1, 0, 0, -2, ...].
For n>0, n in A028957 equivalent to a(n) nonzero. If a(n) nonzero, a(n) = A082451(n) and a(n) = A121362(n).
a(n) = (A082451(n) + A121362(n))/2.
G.f.: x * Product_{k>0} (1 - x^(3*k)) * (1 - x^(5*k)) * (1 + x^(6*k))^2 * (1 + x^(10*k))^2.

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).
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.