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.

A138741 Expansion of q^(-1/2) * eta(q)^3 * eta(q^4) * eta(q^12) / (eta(q^2)^2 * eta(q^3)) in powers of q (unsigned).

Original entry on oeis.org

1, 3, 2, 0, 1, 0, 2, 6, 2, 0, 0, 0, 3, 3, 2, 0, 0, 0, 2, 6, 2, 0, 2, 0, 1, 6, 2, 0, 0, 0, 2, 0, 4, 0, 0, 0, 2, 9, 0, 0, 1, 0, 4, 6, 2, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 6, 2, 0, 2, 0, 1, 6, 4, 0, 0, 0, 0, 6, 2, 0, 0, 0, 4, 3, 2, 0, 2, 0, 2, 6, 0, 0, 0, 0, 3, 0, 2
Offset: 0

Views

Author

Michael Somos, Mar 27 2008

Keywords

Comments

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

Examples

			G.f. = 1 + 3*x + 2*x^2 + x^4 + 2*x^6 + 6*x^7 + 2*x^8 + 3*x^12 + 3*x^13 + ...
G.f. = q + 3*q^3 + 2*q^5 + q^9 + 2*q^13 + 6*q^15 + 2*q^17 + 3*q^25 + 3*q^27 + ...

Links

Crossrefs

Cf. A002175, A008441, A019669, A116604, A121444, A132003.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, (-1)^Quotient[#, 6] {1, 0, 2, 0, 1, 0}[[Mod[#, 6, 1]]] &]]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := SeriesCoefficient[ x^(-1/2) (EllipticTheta[ 2, 0, x]^2 + 3 EllipticTheta[ 2, 0, x^3]^2) / 4, {x, 0, n}]; (* Michael Somos, Sep 08 2015 *)
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # < 3, 1, # == 3, 2 - (-1)^#2, Mod[#, 12] < 6, #2 + 1, True, 1 - Mod[#2, 2]] & @@@ FactorInteger[2 n + 1])]; (* Michael Somos, Sep 08 2015 *)
QP = QPochhammer; s = QP[q^2]^7*QP[q^3]*QP[q^12]^2 / (QP[q]^3*QP[q^4]^2* QP[q^6]^3) + O[q]^90; CoefficientList[s, q] (* Jean-François Alcover, Nov 24 2015 *)
  • PARI
    {a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, (-1)^(d\6) * [0, 1, 0, 2, 0, 1][d%6 + 1]))};
  • 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==2, 0, p==3, 2 - (-1)^e, p%12<6, e+1, 1-e%2 )))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^7 * eta(x^3 + A) * eta(x^12 + A)^2 / (eta(x + A)^3 * eta(x^4 + A)^2 * eta(x^6 + A)^3), n))};

Formula

Expansion of q^(-1/2) * (theta_2(q)^2 + 3 * theta_2(q^3)^2) / 4 in powers of q.
Expansion of phi(q) * psi(q) * psi(q^3) / phi(q^3) in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 12 sequence [ 3, -4, 2, -2, 3, -2, 3, -2, 2, -4, 3, -2, ...].
Moebius transform is period 24 sequence [ 1, -1, 2, 0, 1, -2, -1, 0, -2, -1, -1, 0, 1, 1, 2, 0, 1, 2, -1, 0, -2, 1, -1, 0, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1 + (-1)^e, b(p^e) = e+1 if p == 1, 5 (mod 12), b(p^e) = (1+(-1)^e)/2 if p = 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 6 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A132003.
a(6*n + 3) = a(6*n + 5) = 0.
a(n) = (-1)^n * A116604(n). a(2*n) = A008441(n).
a(6*n) = A002175(n). a(6*n + 1) = 3 * A008441(n). a(6*n + 2) = 2 * A121444(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/2 (A019669). - Amiram Eldar, Dec 28 2023

A122857 Expansion of (phi(q)^2 + phi(q^3)^2) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Sep 14 2006

Keywords

Comments

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

Examples

			G.f. = 1 + 2*q + 2*q^2 + 2*q^3 + 2*q^4 + 4*q^5 + 2*q^6 + 2*q^8 + 2*q^9 + 4*q^10 + ...

References

  • B. C. Berndt, Ramanujan's Notebooks Part IV, Springer-Verlag, see p. 197, Entry 44.

Links

Crossrefs

Cf. A002654, A008441, A019693, A035154, A113446, A125079, A125061, A132003.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, Boole[n == 0], 2 DivisorSum[ n, KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Jul 09 2013 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q]^2 + EllipticTheta[ 3, 0, q^3]^2) / 2, {q, 0, n}]; (* Michael Somos, Jul 09 2013 *)
  • PARI
    {a(n) = if( n<1, n==0, 2 * sumdiv( n, d, kronecker( -36, d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); 2 * prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p<5, 1, p%12<6, e+1, !(e%2) )))};
  • PARI
    {a(n) = my(A, p, e); if( n<1, n==0, A = factor(n); prod( k=1, matsize(A)[1],
     [p, e] = A[k, ]; if( p==2, 1, p==3, 1+e%2*2, p%4==1, e+1, !(e%2) )))};

Formula

Expansion of eta(q^2)^3 * eta(q^3)^2 * eta(q^6) / (eta(q)^2 * eta(q^4)* eta(q^12)) in powers of q.
Expansion of 2 * psi(q) * psi(q^2) * psi(q^3) / psi(q^6) - phi(q^3)^2 in powers of q. - Michael Somos, Jul 09 2013
Euler transform of period 12 sequence [ 2, -1, 0, 0, 2, -4, 2, 0, 0, -1, 2, -2, ...].
Moebius transform is period 12 sequence [ 2, 0, 0, 0, 2, 0, -2, 0, 0, 0, -2, 0, ...].
a(12*n + 7) = a(12*n + 11) = 0.
a(n) = 2 * b(n) where b(n) is multiplicative and b(2^e) = b(3^e) = 1, b(p^e) = e+1 if p == 1, 5 (mod 12), a(p^e) == (1-(-1)^e)/2 if p == 7, 11 (mod 12).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 4 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A125061.
A035154(n) = a(n) / 2 if n > 0. A008441(n) = a(4*n + 1) / 2. A125079(n) = a(2*n + 1) / 2. A113446(3*n + 1) = A002654(3*n + 1) = a(3*n + 1) / 2.
a(n) = (-1)^n * A132003(n). Expansion of (phi(-q^3) / phi(-q)) * phi(-q^2) * phi(-q^6) in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 05 2023
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/3 = 2.0943951... (A019693). - Amiram Eldar, Nov 21 2023

A132004 Expansion of (1 - phi(q^3) / phi(q) * phi(-q^2) * phi(-q^6)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Aug 06 2007

Keywords

Comments

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

Examples

			G.f. = x - x^2 + x^3 - x^4 + 2*x^5 - x^6 - x^8 + x^9 - 2*x^10 - x^12 + 2*x^13 + ...

References

  • N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Equation (32.72).

Links

Crossrefs

Cf. A008441, A035154, A053692, A122856, A122865, A125079, A132003, A258277, A258278.

Programs

  • Mathematica
    a[ n_] := If[ n < 1, 0, DivisorSum[ n, (-1)^(n + #) KroneckerSymbol[ -36, #] &]]; (* Michael Somos, Nov 01 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # < 5, -(-1)^#, Mod[#, 4] == 3, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Nov 01 2015 *)
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, (-1)^(n+d) * kronecker( -36, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (1 - eta(x + A)^2 * eta(x^4 + A) * eta(x^6 + A)^7 / (eta(x^2 + A)^3 * eta(x^3 + A)^2 * eta(x^12 + A)^3)) / 2, n))};
  • 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==3, 1, p==2, -1, p%4==1, e+1, 1-e%2)))};

Formula

Expansion of (1 - eta(q)^2 * eta(q^4) * eta(q^6)^7 / (eta(q^2)^3 * eta(q^3)^2 * eta(q^12)^3)) / 2 in powers of q.
a(n) is multiplicative with a(2^e) = 2*0^e - 1, a(3^e) = 1, a(p^e) = e + 1 if p == 1 (mod 4), a(p^e) = (1 + (-1)^e) / 2 if p == 3 (mod 4).
G.f.: Sum_{k>0} x^k / (1 + x^k) * Kronecker(-36, k).
a(3*n) = a(n). -2 * a(n) = A132003(n) unless n = 0. a(2*n) = - A035154(n). a(2*n + 1) = A125079(n).
a(n) = (-1)^n * A035154(n). a(12*n + 7) = a(12*n + 11) = 0. - Michael Somos, Nov 01 2015
a(3*n + 1) = A258277(n). a(3*n + 2) = - A258278(n). a(4*n + 1) = A008441(n). a(4*n + 2) = - A125079(n). - Michael Somos, Nov 01 2015
a(6*n) = - A035154(n). a(6*n + 2) = - A122865(n). a(6*n + 4) = - A122856(n). - Michael Somos, Nov 01 2015
a(8*n + 1) = A113407(n). a(8*n + 5) = 2 * A053692(n). - Michael Somos, Nov 01 2015
Showing 1-3 of 3 results.

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