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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A134013 Expansion of q * phi(q) * psi(q^8) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 02 2007

Keywords

Comments

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

Examples

			G.f. = q + 2*q^2 + 2*q^5 + q^9 + 4*q^10 + 2*q^13 + 2*q^17 + 2*q^18 + 3*q^25 + ...

Links

Crossrefs

Cf. A003881, A008441, A053692, A112301, A113407, A134014.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/2) EllipticTheta[ 3, 0, q] EllipticTheta[ 2, 0, q^4], {q, 0, n}]; (* Michael Somos, Oct 30 2015 *)
  • PARI
    {a(n) = if( n>0 && (n+1)%4\2, (n%4) * sumdiv( n/gcd(n,2), d, (-1)^(d\2)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^16 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^8 + A)), n))};

Formula

Expansion of eta(q^2)^5 * eta(q^16)^2 / ( eta(q)^2 * eta(q^4)^2 * eta(q^8) ) in powers of q.
Euler transform of period 16 sequence [ 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, ...].
Moebius transform is period 16 sequence [ 1, 1, -1, -2, 1, -1, -1, 0, 1, 1, -1, 2, 1, -1, -1, 0, ...].
a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if 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. is a period 1 Fourier series which satisfies f(-1 / (16 t)) = 2 (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A134014.
a(4*n) = a(4*n + 3) = a(8*n + 6) = 0. a(8*n + 2) = 2 * a(4*n + 1).
G.f.: Sum_{k>0} Kronecker(-4, k) * x^k * (1 + x^k)^2 / (1 - x^(4*k)).
a(n) = -(-1)^n * A112301(n). a(4*n + 1) = A008441(n). a(8*n + 1) = A113407(n). a(8*n = 5) = 2 * A053692(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 (A003881). - Amiram Eldar, Nov 24 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

A246862 Expansion of phi(x) * f(x^3, x^5) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 1, 4, 1, 2, 2, 0, 4, 0, 0, 2, 0, 3, 2, 2, 0, 3, 4, 0, 2, 2, 2, 0, 2, 0, 2, 2, 0, 4, 0, 0, 1, 4, 0, 2, 2, 0, 5, 2, 2, 2, 4, 0, 0, 0, 0, 2, 4, 2, 0, 2, 0, 4, 2, 0, 0, 2, 0, 1, 2, 0, 2, 6, 0, 0, 4, 1, 8, 0, 0, 2, 0, 0, 2, 2, 2, 2, 0, 0, 2, 4, 0, 4, 2, 2
Offset: 0

Views

Author

Michael Somos, Sep 05 2014

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^3 + 4*x^4 + x^5 + 2*x^6 + 2*x^7 + 4*x^9 + 2*x^12 + ...
G.f. = q + 2*q^17 + q^49 + 4*q^65 + q^81 + 2*q^97 + 2*q^113 + 4*q^145 + ...

Links

Crossrefs

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741, A214264, A226192, A246863.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^3, x^8] QPochhammer[ -x^5, x^8] QPochhammer[ x^8], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, issquare(16 * n + 1) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 1)))};

Formula

Euler transform of period 16 sequence [ 2, -3, 3, -1, 3, -4, 2, -2, 2, -4, 3, -1, 3, -3, 2, -2, ...].
Convolution of A000122 and A214264.
a(9*n + 2) = a(9*n + 8) = 0. a(9*n + 5) = A246863(n).
a(n) = A113407(2*n) = A226192(2*n) = A008441(4*n) = A134343(4*n) = A116604(8*n) = A125079(8*n) = A129447(8*n) = A138741(8*n).

A246863 Expansion of phi(x) * f(x^1, x^7) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Sep 05 2014

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 + 2*x^4 + 2*x^5 + x^7 + 2*x^8 + 2*x^9 + 3*x^10 + ...
G.f. = q^9 + 3*q^25 + 2*q^41 + 2*q^73 + 2*q^89 + q^121 + 2*q^137 + 2*q^153 + ...

Links

Crossrefs

Cf. A000122, A008441, A113407, A116604, A125079, A129447, A134343, A138741,A214263, A226192, A246862.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] QPochhammer[ -x^1, x^8] QPochhammer[ -x^7, x^8] QPochhammer[ x^8], {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, issquare(16 * n + 9) + 2 * sum(i=1, sqrtint(n), issquare(16 * (n - i^2) + 9)))};

Formula

Euler transform of period 16 sequence [ 3, -4, 2, -1, 2, -3, 3, -2, 3, -3, 2, -1, 2, -4, 3, -2, ...].
Convolution of A000122 and A214263.
a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = A246862(n).
a(n) = A113407(2*n + 1) = - A226192(2*n + 1) = A008441(4*n + 2) = A134343(4*n + 2) = A116604(8*n + 4) = A125079(8*n + 4) = A129447(8*n + 4) = A138741(8*n + 4).
Previous Showing 11-14 of 14 results.

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