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 1-9 of 9 results.

A128580 Expansion of phi(x^3) * psi(x^4) - x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Mar 11 2007

Keywords

Comments

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

Examples

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

Links

Crossrefs

Cf. A113780, A115660, A128581, A128582, A259668.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^2] - EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jul 12 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker(-12, d) * kronecker(2, n/d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^2 + A) * eta(x^8 + A) * eta(x^12 + A)^3 / (eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^24 + A)), n))};

Formula

a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1)* (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1+(-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ -1, -2, 0, 0, -1, -1, -1, -1, 0, -2, -1, -2, -1, -2, 0, -1, -1, -1, -1, 0, 0, -2, -1, -2, ...].
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11)= 0.
G.f.: Product_{k>0} (1 - x^(8*k)) * (1 - x^(12*k))^2 / ((1 + x^k) * (1 + x^(2*k))^2 * (1 - x^(3*k)) * (1 + x^(12*k))).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (x^k - x^(3*k)) / (1 + x^(4*k)) * Kronecker(-12, k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(2*k) + x^(4*k)) * Kronecker(2, k).
a(n) = A128581(2*n + 1) = A115660(2*n + 1). a(3*n + 2) = -2 * A128582(n). a(12*n) = A113780(n).
a(2*n) = A259668(n). a(3*n + 1) = - A128580(n). - Michael Somos, Jul 12 2015

A129402 Expansion of phi(x^3) * psi(x^4) + x * phi(x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 2, 2, 1, 2, 0, 2, 0, 0, 2, 0, 3, 1, 2, 2, 2, 4, 0, 0, 0, 0, 2, 0, 3, 0, 2, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 3, 4, 2, 1, 2, 0, 2, 0, 0, 2, 0, 2, 2, 2, 2, 4, 2, 0, 0, 0, 0, 0, 0, 3, 0, 4, 2, 0, 2, 0, 2, 0, 0, 0, 0, 4, 3, 2, 2, 0, 4, 0, 2, 0, 0, 4, 0, 1, 0, 2
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 + x + 2*x^2 + 2*x^3 + x^4 + 2*x^5 + 2*x^7 + 2*x^10 + 3*x^12 + x^13 + 2*x^14 + ...
G.f. = q + q^3 + 2*q^5 + 2*q^7 + q^9 + 2*q^11 + 2*q^15 + 2*q^21 + 3*q^25 + q^27 + ...

References

  • Nathan J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 83, Eq. (32.57).

Links

Crossrefs

Cf. A000377, A128580, A128582, A113780, A190615, A257920, A259895, A260308, A261115, A261118, A261119, A261122.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Nov 11 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ x^3] QPochhammer[ -x, x] QPochhammer[ x^6, -x^6], {x, 0, n}]; (* Michael Somos, Nov 11 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n+1; sumdiv( n, d, kronecker( -6, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^3 * eta(x^6 + A) * eta(x^24 + A) / (eta(x + A) * eta(x^8 + A) * eta(x^12 + A)^2), n))};

Formula

Expansion of f(x^2) * f(-x^3) / (chi(-x) * chi(x^6)) in powers of x where f(), chi() are Ramanujan theta functions.
Expansion of q^(-1/2) * eta(q^3) * eta(x^4)^3 * eta(q^6) * eta(q^24) / (eta(q) * eta(q^8) * eta(q^12)^12) in powers of q.
Euler transform of period 24 sequence [ 1, 1, 0, -2, 1, -1, 1, -1, 0, 1, 1, -2, 1, 1, 0, -1, 1, -1, 1, -2, 0, 1, 1, -2, ...].
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 5, 7, 11 (mod 24), b(p^e) = (1 + (-1)^e)/2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (48 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is g.f. for A190611.
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0. a(3*n + 1) = a(n).
a(n) = A000377(2*n + 1). a(3*n + 2) = 2 * A128582(n). a(12*n) = A113780(n).
a(n) = (-1)^n * A190615(n) = (-1)^floor( (n+1) / 2) * A128580(n). - Michael Somos, Nov 11 2015
a(2*n) = A261118(n). a(2*n + 1) = A261119(n). a(3*n) = A261115(n). - Michael Somos, Nov 11 2015
a(4*n) = A260308(n). a(4*n + 1) = A257920(n). a(4*n + 2) = 2 * A259895(n). - Michael Somos, Nov 11 2015
a(n) = - A261122(4*n + 2). - Michael Somos, Nov 11 2015
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A134177 Expansion of phi(-x^3) * psi(x^4) + x * phi(-x) * psi(x^12) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 11 2007

Keywords

Comments

Number 64 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Examples

			G.f. = 1 + x - 2*x^2 - 2*x^3 + x^4 + 2*x^5 - 2*x^7 - 2*x^10 + 3*x^12 + x^13 - ...
G.f. = q + q^3 - 2*q^5 - 2*q^7 + q^9 + 2*q^11 - 2*q^15 - 2*q^21 + 3*q^25 + q^27 + ...

Links

Crossrefs

Cf. A113780, A128580.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 2 n + 1}, DivisorSum[ m, KroneckerSymbol[ 12, #] KroneckerSymbol[ -2, m/#] &]]]; (* Michael Somos, Jun 24 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 4, 0, x^3] EllipticTheta[ 2, 0, x^2] + EllipticTheta[ 4, 0, x] EllipticTheta[ 2, 0, x^6]) / (2 x^(1/2)), {x, 0, n}]; (* Michael Somos, Jun 24 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 2*n + 1; sumdiv(n, d, kronecker( 12, d) * kronecker( -2, n/d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^4 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^4 / ( eta(x + A) * eta(x^4 + A)^3 * eta(x^6 + A)^3* eta(x^24 + A) ), n))};
  • 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, 1, p%24>12, !(e%2), (e+1) * kronecker(3, p)^e)))};

Formula

Expansion of q^(-1/2) * eta(q^2)^4 * eta(q^3) * eta(q^8) * eta(q^12)^4 / ( eta(q) * eta(q^4)^3 * eta(q^6)^3 * eta(q^24) ) in powers of q.
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = 1, b(p^e) = e+1 if p == 1, 11 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 7 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
Euler transform of period 24 sequence [ 1, -3, 0, 0, 1, -1, 1, -1, 0, -3, 1, -2, 1, -3, 0, -1, 1, -1, 1, 0, 0, -3, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (96 t)) = 96^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
G.f.: Sum_{k>=0} a(k) * x^(2*k+1) = Sum_{k>0} Kronecker( -2, k) * (x^k - x^(3*k)) / (1 - x^(2*k) + x^(4*k)).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 - x^(3*k)) * (1 + x^(6*k)) * (1 - x^(2*k) + x^(4*k))^2 / (1 - x^(4*k) + x^(8*k)).
a(n) = (-1)^n * A128580(n). a(12*n) = A113780(n).

A259668 Expansion of psi(-x)^2 * psi(x^3)^2 / (phi(-x^4) * psi(-x^6)) in power of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 02 2015

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^5 + 3*x^6 - 2*x^7 + 2*x^8 - 2*x^11 + 3*x^12 + ...
G.f. = q - 2*q^5 + q^9 - 2*q^21 + 3*q^25 - 2*q^29 + 2*q^33 - 2*q^45 + ...

Links

Crossrefs

Cf. A113780, A115660, A128580, A128581, A134177.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ x^8] QPochhammer[x^12, x^24] QPochhammer[ x^6] (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6])^2, {x, 0, n}];
a[ n_] := SeriesCoefficient[ (EllipticTheta[2, 0, x^(3/2)] EllipticTheta[ 2, Pi/4, x^(1/2)])^2 / (2^(5/2) x^(1/4) EllipticTheta[ 4, 0, x^4] EllipticTheta[ 2, Pi/4, x^3]), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^6 + A)^3 * eta(x^8 + A) * eta(x^12 + A) / (eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^24 + A)), n))};

Formula

Expansion of f(-x^8) * f(-x, -x^5)^2 / psi(-x^6) in powers of x where psi(), f() are Ramanujan theta functions.
Euler transform of period 24 sequence [ -2, 0, 0, 0, -2, -1, -2, -1, 0, 0, -2, -2, -2, 0, 0, -1, -2, -1, -2, 0, 0, 0, -2, -2, ...].
a(n) = A128580(2*n) = A134177(2*n) = A115660(4*n) = A128581(4*n).
a(6*n + 1) = -2 * A113780(n).

A261115 Expansion of f(x, x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Aug 08 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 2*x + 3*x^4 + 2*x^5 + 3*x^8 + 4*x^9 + 2*x^12 + 2*x^13 + 2*x^16 + ...
G.f. = q + 2*q^7 + 3*q^25 + 2*q^31 + 3*q^49 + 4*q^55 + 2*q^73 + 2*q^79 + ...

Links

Crossrefs

Cf. A113780, A115660, A128580, A128581, A129402, A192013, A260110.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x + A)^2 * eta(x^4 + A)^3 * eta(x^24 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q^2)^5 * eta(q^8) * eta(q^12)^2 / (eta(q)^2 * eta(q^4)^3 * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ 2, -3, 2, 0, 2, -3, 2, -1, 2, -3, 2, -2, 2, -3, 2, -1, 2, -3, 2, 0, 2, -3, 2, -2, ...].
a(n) = (-1)^n * A260110(n) = A128580(3*n) = A129402(3*n) = A115660(6*n + 1) = A128581(6*n + 1) = A192013(6*n + 1).
a(4*n) = A113780(n). a(4*n + 1) = 2 * A260089(n). a(4*n + 2) = a(4*n + 3) = 0.

A263571 Expansion of f(x^2, x^2) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Oct 21 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + 2*x^2 + 2*x^3 + x^5 + 2*x^7 + 3*x^8 + 2*x^9 + 2*x^10 + ...
G.f. = q + q^4 + 2*q^7 + 2*q^10 + q^16 + 2*q^22 + 3*q^25 + 2*q^28 + ...

Links

Crossrefs

Cf. A113780, A115660, A128581, A128582, A192013, A260089, A261115, A263548, A263577.
Cf. A000122, A000700, A010054, A121373.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, m/d], {d, Divisors[ m]}]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] EllipticTheta[ 3, 0, x^2] EllipticTheta[ 2, Pi/4, x^(3/2)] / (2^(1/2) x^(3/8)), {x, 0, n}];
  • PARI
    {a(n) = my(m); if( n<0, 0, m = 3*n + 1; sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^3 + A) * eta(x^4 + A)^4 * eta(x^12 + A) / (eta(x + A) * eta(x^8 + A)^2), n))};

Formula

Expansion of chi(x) * phi(x^2) * psi(-x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q^3) * eta(q^4)^4 * eta(q^12) / (eta(q) * eta(q^6) * eta(q^8)^2) in powers of q.
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 24^(1/2) (t/i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A263577.
a(n) = A115660(3*n + 1) = A192013(3*n + 1) = A128581(6*n + 2).
a(2*n) = A261115(n). a(2*n + 1) = A263548(n). a(4*n + 1) = a(n). a(4*n + 3) = 2 * A128582(n).
a(8*n + 4) = a(8*n + 6) = 0. a(8*n) = A113780(n). a(8*n + 2) = 2 * A260089(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(6) = 1.282549... . - Amiram Eldar, Dec 28 2023

A114913 Numbers k such that A114912(k) = 1. Numbers k such that A000009(k) == 2 (mod 4).

Original entry on oeis.org

3, 4, 8, 10, 13, 14, 17, 18, 19, 24, 25, 28, 32, 39, 42, 43, 47, 48, 50, 52, 54, 55, 62, 67, 69, 73, 74, 75, 76, 78, 83, 84, 87, 88, 89, 90, 95, 99, 101, 103, 105, 108, 109, 112, 113, 118, 119, 123, 125, 127, 130, 132, 134, 138, 140, 143, 144, 147, 149, 153, 154, 157
Offset: 1

Views

Author

Christian G. Bower, Jan 06 2006

Keywords

Comments

All the terms are the sum of a generalized pentagonal number A001318 and a square A000290.
Let 24*k+1 = p_1^e_1 * ... * p_r^e_r * q_1^f_1 * ... * q_s^f_s, where the p_i's are distinct primes == 1, 5, 7, or 11 (mod 24) and the q_i's are distinct primes == 13, 17, 19, or 23 (mod 24). Then k belongs to the sequence iff all of the f_i's are even and all but one of the e_i's are even and the one e_i which is odd is == 1 (mod 4). - Dean Hickerson, Jan 19 2006

Links

Crossrefs

Cf. A000009, A000290, A001318, A114912, A114914.
A111174 is a subsequence.
See comments in A113780 for explanation.

Programs

A260110 Expansion of f(-x, -x) * f(x^4, x^8) in powers of x where f(,) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 16 2015

Keywords

Comments

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

Examples

			G.f. = 1 - 2*x + 3*x^4 - 2*x^5 + 3*x^8 - 4*x^9 + 2*x^12 - 2*x^13 + 2*x^16 + ...
G.f. = q - 2*q^7 + 3*q^25 - 2*q^31 + 3*q^49 - 4*q^55 + 2*q^73 - 2*q^79 + ...

Links

Crossrefs

Cf. A134177, A190615, A229723, A260089.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] EllipticTheta[ 4, 0, x^12] / QPochhammer[ x^4, x^8], {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^8 + A) * eta(x^12 + A)^2 / (eta(x^2 + A) * eta(x^4 + A) * eta(x^24 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q)^2 * eta(q^8) * eta(q^12)^2 / (eta(q^2) * eta(q^4) * eta(q^24)) in powers of q.
Euler transform of period 24 sequence [ -2, -1, -2, 0, -2, -1, -2, -1, -2, -1, -2, -2, -2, -1, -2, -1, -2, -1, -2, 0, -2, -1, -2, -2, ...].
a(n) = A134177(3*n) = A190615(3*n) = A229723(6*n + 1). a(4*n + 2) = a(4*n + 3) = 0. a(4*n) = A113780(n). a(4*n + 1) = -2 * A260089(n).

A279947 Expansion of f(x^2, x^2) * f(-x, -x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Dec 23 2016

Keywords

Comments

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

Examples

			G.f. = 1 - x + 2*x^2 - 2*x^3 - x^5 - 2*x^7 + 3*x^8 - 2*x^9 + 2*x^10 + ...
G.f. = q - q^4 + 2*q^7 - 2*q^10 - q^16 - 2*q^22 + 3*q^25 - 2*q^28 + ...

Links

Crossrefs

Cf. A113780, A128581, A128582, A128583, A128591, A190611, A260089, A261115, A261122, A263548, A263571.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 3 n + 1}, (-1)^n DivisorSum[ m, KroneckerSymbol[ 2, #] KroneckerSymbol[ -3, m/#] &]]];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] QPochhammer[ x^6] QPochhammer[ x, x^6] QPochhammer[ x^5, x^6], {x, 0, n}];
  • PARI
    {a(n) = my(m); if( n<0, 0, m = 3*n + 1; (-1)^n * sumdiv( m, d, kronecker( 2, d) * kronecker( -3, m/d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^5 * eta(x^6 + A)^2 / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A)^2), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(3*n + 1); prod( k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, -(-1)^e, p==3, (-1)^e, p%24==1 || p%24==7, e+1, p%24==5 || p%24==11, (e+1)*(-1)^e, !(e%2))))};

Formula

Expansion of chi(-x) * phi(x^2) * psi(x^3) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-1/3) * eta(q) * eta(q^4)^5 * eta(q^6)^2 / (eta(q^2)^3 * eta(q^3) * eta(q^8)^2) in powers of q.
Euler transform of period 24 sequence [ -1, 2, 0, -3, -1, 1, -1, -1, 0, 2, -1, -4, -1, 2, 0, -1, -1, 1, -1, -3, 0, 2, -1, -2, ...].
a(n) = b(3*n + 1) where b() is multiplicative with b(2^e) = -(-1)^e if e>0, b(3^e) = (-1)^e, b(p^e) = e+1 if p == 1, 7 (mod 24), b(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), b(p^e) = (1 + (-1)^e) / 2 if p == 13, 17, 19, 23 (mod 24).
a(n) = (-1)^n * A263571(n) = A128581(3*n + 1) = - A190611(3*n + 1) = - A261122(6*n + 2).
a(2*n) = A261115(n).
a(2*n + 1) = - A263548(n).
a(8*n + 4) = a(8*n + 6) = 0.
a(4*n + 1) = -a(n).
a(4*n + 3) = -2 * A128582(n).
a(8*n) = A113780(n).
a(8*n + 2) = 2 * A260089(n).
a(16*n + 3) = -2 * A128583(n).
a(16*n + 7) = -2 * A128591(n).
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