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.

A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m = -20.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2, 4, 0, 2, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 1, 2, 0, 2, 0, 4
Offset: 1

Views

Author

N. J. A. Sloane

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -20. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

Examples

			q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.

Links

Crossrefs

Cf. A028586, A033718, A033764, A111949, A124233, A129390.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

Programs

  • Mathematica
    QP = QPochhammer; s = (1/q) * (QP[q^2]*QP[q^4]*QP[q^5]*(QP[q^10] / (QP[q]* QP[q^20]))-1) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 04 2015 *)
a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Dec 12 2017 *)
  • PARI
    direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))
  • PARI
    {a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ Michael Somos, Sep 10 2005
  • PARI
    {a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -20, p) * X) )[n])} \\ Michael Somos, Sep 10 2005
  • PARI
    {a(n) = if( n<1, 0, qfrep([1, 0; 0, 5], n)[n] + qfrep([2, 1; 1, 3], n)[n])} \\ Michael Somos, Oct 21 2006

Formula

Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - Michael Somos, Sep 10 2005
G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - Michael Somos, Sep 10 2005
a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.
Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - Michael Somos, Oct 21 2006
Expansion of -1 + (phi(q) * phi(q^5) + phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4)* psi(q^20)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions.
2*a(n) = A028586(n) + A033718(n) if n>0. - Michael Somos, Oct 21 2006
a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Oct 11 2022

A129391 Expansion of phi(-x) * phi(x^5) / (chi(-x^2) * chi(-x^10)) in powers of x where phi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 1, -2, 3, 0, 0, -2, 0, 0, 4, -2, 1, -4, 2, 0, 0, -2, 0, 0, 2, -2, 3, -2, 3, 0, 0, 0, 0, 0, 2, -6, 0, -2, 4, 0, 0, -2, 0, 0, 5, -2, 0, -4, 2, 0, 0, 0, 0, 0, 2, -2, 4, -2, 2, 0, 0, -2, 0, 0, 1, -4, 1, -2, 4, 0, 0, -4, 0, 0, 4, 0, 2, -6, 2, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Michael Somos, Apr 13 2007

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + x^2 - 2*x^3 + 3*x^4 - 2*x^7 + 4*x^10 - 2*x^11 + x^12 - 4*x^13 + ...
G.f. = q - 2*q^3 + q^5 - 2*q^7 + 3*q^9 - 2*q^15 + 4*q^21 - 2*q^23 + q^25 - ...

Links

Crossrefs

Cf. A129390, A143323.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -20, #]&]]; (* Michael Somos, Nov 12 2015 *)
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv(2*n + 1, d, kronecker( -20, d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==5, 1, p%20 <10, (-1)^( ((p%20)%4 == 3)*e) * (e+1), 1-e%2 )))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A) * eta(x^10 + A)^4 / (eta(x^2 + A)^2 * eta(x^5 + A)^2 * eta(x^20 + A)), n))};

Formula

Expansion of q^(-1/2) * eta(q)^2 * eta(q^4) * eta(q^10)^4 / (eta(q^2)^2 * eta(q^5)^2 * eta(q^20)) in powers of q.
Euler transform of period 20 sequence [ -2, 0, -2, -1, 0, 0, -2, -1, -2, -2, -2, -1, -2, 0, 0, -1, -2, 0, -2, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0, b(5^e) = 1, b(p^e) = (-1)^e* (e+1) if p == 3, 7 (mod 20), b(p^e) = e+1 if p == 1, 9 (mod 20), b(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20).
G.f.: Sum_{k>0} a(k) * x^(2*k - 1) = Sum_{k>0} (-1)^k * f(x^(2*k - 1)) where f(x) := x * (1 - x^2) * (1 - x^6) / (1 - x^10).
a(n) = (-1)^n * A129390(n).
a(n) = A111949(2*n + 1) = A143323(2*n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (40 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A143323.

A143323 Expansion of eta(q^2)^4 * eta(q^5) * eta(q^20)^2 / ( eta(q) * eta(q^4)^2 * eta(q^10)^2 ) in powers of q.

Original entry on oeis.org

1, 1, -2, -1, 1, -2, -2, 1, 3, 1, 0, 2, 0, -2, -2, -1, 0, 3, 0, -1, 4, 0, -2, -2, 1, 0, -4, 2, 2, -2, 0, 1, 0, 0, -2, -3, 0, 0, 0, 1, 2, 4, -2, 0, 3, -2, -2, 2, 3, 1, 0, 0, 0, -4, 0, -2, 0, 2, 0, 2, 2, 0, -6, -1, 0, 0, -2, 0, 4, -2, 0, 3, 0, 0, -2, 0, 0, 0, 0, -1, 5, 2, -2, -4, 0, -2, -4, 0, 2, 3, 0, 2, 0, -2, 0, -2, 0, 3, 0, -1, 2, 0, -2, 0, 4
Offset: 1

Views

Author

Michael Somos, Aug 07 2008

Keywords

Comments

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

Examples

			G.f. = q + q^2 - 2*q^3 - q^4 + q^5 - 2*q^6 - 2*q^7 + q^8 + 3*q^9 + q^10 + ...

Links

Crossrefs

Cf. A111949, A129391.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 2 q^(13/8) EllipticTheta[ 4, 0, q^2]^2 QPochhammer[ q^20]^2 / ( QPochhammer[ q] EllipticTheta[ 2, 0, q^(5/2)]), {q, 0, n}]; (* Michael Somos, Apr 07 2015 *)
  • PARI
    {a(n) = if( n<1, 0, (-1)^n * (qfrep([2, 1; 1, 3], n)[n] - qfrep([1, 0; 0, 5], n)[n] ))};
  • PARI
    {a(n) = local(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^5 + A) * eta(x^20 + A)^2 / ( eta(x + A) * eta(x^4 + A)^2 * eta(x^10 + A)^2 ), n))};

Formula

Expansion of q * phi(-q^2) * chi(q) * psi(q^10) * chi(-q^5) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q * phi(-q^2)^2 * psi(-q^5)^2 / (f(-q) * f(-q^5)) in powers of q where phi(), psi(), f() are Ramanujan theta functions. - Michael Somos, Apr 07 2015
Euler transform of period 20 sequence [ 1, -3, 1, -1, 0, -3, 1, -1, 1, -2, 1, -1, 1, -3, 0, -1, 1, -3, 1, -2, ...].
Multiplicative with a(2^e) = -(-1)^e unless e=0, a(p^e) = 1 if p=5, a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20), a(p^e) = e+1 if p == 1, 9 (mod 20), a(p^e) = (e+1)(-1)^e if p == 3, 7 (mod 20).
G.f. is a period 1 Fourier series which satisfies f(-1 / (20 t)) = 20^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A129391.
G.f.: Sum_{k>0} -(-1)^k F(x^(2*k - 1)) where F(x) = x * (1 + x) * (1 - x^2) / (1 + x^5).
G.f.: x * Product_{k>0} (1 - x^k) * (1 + x^(2*k-1))^2 * (1 - x^(5*k)) * ( 1 + x^(10*k))^2.
a(n) = -(-1)^n * A111949(n).
Showing 1-3 of 3 results.

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

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