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 1-8 of 8 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

A115660 Expansion of (phi(q) * phi(q^6) - phi(q^2) * phi(q^3)) / 2 in powers of q where phi() is a Ramanujan theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jan 28 2006

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Number 41 of the 74 eta-quotients listed in Table I of Martin (1996). - Michael Somos, Mar 14 2012

Examples

			G.f. = q - q^2 - q^3 + q^4 - 2*q^5 + q^6 + 2*q^7 - q^8 + q^9 + 2*q^10 - 2*q^11 + ...

Links

Crossrefs

Cf. A000377, A046113, A108563, A128580, A128581, A192013, A259668, A260308, A261115, A263548, A263571.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ q QPochhammer[ q] QPochhammer[ q^4] QPochhammer[ q^6] QPochhammer[ q^24] / (QPochhammer[ q^3] QPochhammer[ q^8]), {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^6] - EllipticTheta[ 3, 0, q^2] EllipticTheta[ 3, 0, q^3]) / 2, {q, 0, n}]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ 2, d] KroneckerSymbol[ -3, n/d], {d, Divisors[ n]}]]; (* Michael Somos, Apr 19 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # == 1, 1, # < 5, (-1)^#2, Mod[#, 24] < 12, (#2 + 1) KroneckerSymbol[ #, 12]^#2, True, 1 - Mod[#2, 2]]& @@@ FactorInteger[n])]; (* Michael Somos, Oct 22 2015 *)
  • 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<5, (-1)^e, p%24<12, (e+1) * kronecker( p, 12)^e, 1-e%2)))};
  • PARI
    {a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A) * eta(x^6 + A) * eta(x^24 + A) / (eta(x^3 + A) * eta(x^8 + A)), n))};
  • PARI
    {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 2, d) * kronecker( -3, n/d)))};

Formula

Expansion of eta(q) * eta(q^4) * eta(q^6) * eta(q^24) / (eta(q^3) * eta(q^8)) 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) is multiplicative with a(2^e) = a(3^e) = (-1)^e, a(p^e) = e+1 if p == 1, 7 (mod 24), a(p^e) = (e+1) * (-1)^e if p == 5, 11 (mod 24), a(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 / (24 t)) = 24^(1/2) (t/i) f(t) where q = exp(2 Pi i t).
G.f.: Sum_{k>0} Kronecker(k,8) * x^k / (1 + x^k + x^(2*k)) = Sum_{k>0} Kronecker(k,3) * x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
abs(a(n)) = A000377(n). a(n) = (-1)^n * A128581(n). a(2*n) = a(3*n) = -a(n). a(2*n + 1) = A128580(n). - Michael Somos, Mar 14 2012
abs(a(n)) = A192013(n) unless n=0. - Michael Somos, Oct 22 2015
a(3*n + 1) = A263571(n). a(4*n) = A259668(n). a(6*n + 1) = A261115(n). a(6*n + 4) = A263548(n). a(8*n + 1) = A260308(n). - Michael Somos, Oct 22 2015
a(n) = A000377(n) - A108563(n) = A046113(n) - A000377(n). - Michael Somos, Oct 22 2015

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).

A260308 Expansion of psi(x) * phi(x^3) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Jul 22 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + 3*x^3 + 2*x^4 + 3*x^6 + 2*x^9 + x^10 + 2*x^12 + 4*x^13 + ...
G.f. = q + q^9 + 3*q^25 + 2*q^33 + 3*q^49 + 2*q^73 + q^81 + 2*q^97 + ...

Links

Crossrefs

Cf. A115660, A128580, A128581, A129402, A134177, A190615, A192013, A259668.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, DivisorSum[ 8 n + 1, KroneckerSymbol[ -6, #] &]];
a[ n_] := If[ n < 0, 0, Times @@ (Which[ # <= 3, Mod[#, 2], Mod[#, 24] > 12, 1 - Mod[#2, 2], True, (#2  + 1) KroneckerSymbol[ 3, #]^#2] & @@@ FactorInteger @ (8 n + 1))];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^3] EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
  • PARI
    {a(n) = if( n<0, 0, sumdiv( 8*n + 1, d, kronecker( -6, d)))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, factor(8*n + 1); 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)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^6 + A)^5 / (eta(x + A) * eta(x^3 + A)^2 * eta(x^12 + A)^2), n))};

Formula

Expansion of q^(-1/8) * eta(q^2)^2 * eta(q^6)^5 / (eta(q) * eta(q^3)^2 * eta(q^12)^2) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 3, -1, 1, -4, 1, -1, 3, -1, 1, -2, ...].
a(n) = A259668(2*n) = A128580(4*n) = A129402(4*n) = A134177(4*n) = A190615(4*n) = A115660(8*n + 1) = A128581(8*n + 1) = A192013(8*n + 1).

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

A258587 Expansion of f(-x, -x) * f(x^2, x^10) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

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

Views

Author

Michael Somos, Nov 06 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^3 + 2*x^4 + 2*x^6 - 2*x^9 + x^10 - 4*x^11 + 2*x^14 + ...
G.f. = q^2 - 2*q^5 + q^8 - 2*q^11 + 2*q^14 + 2*q^20 - 2*q^29 + q^32 - 4*q^35 + ...

Links

Crossrefs

Cf. A128581, A128582, A190611, A263548, A263571.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^12] QPochhammer[ -x^10, x^12] QPochhammer[ x^12], {x, 0, n}];
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] QPochhammer[ -x^2, x^4] EllipticTheta[ 2, 0, x^3] / (2^(1/2) x^(3/4)), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 * eta(x^4 + A)^2 * eta(x^6 + A) * eta(x^24 + A) / (eta(x^2 + A)^2 * eta(x^8 + A) * eta(x^12 + A)), n))};

Formula

Expansion of phi(-x) * chi(x^2) * psi(-x^6) in powers of x where phi(), psi(), chi() are Ramanujan theta functions.
Expansion of q^(-2/3) * eta(q)^2 * eta(q^4)^2 * eta(q^6) * eta(q^24) / (eta(q^2)^2 * eta(q^8) * eta(q^12)) in powers of q.
Euler transform of period 24 sequence [ -2, 0, -2, -2, -2, -1, -2, -1, -2, 0, -2, -2, -2, 0, -2, -1, -2, -1, -2, -2, -2, 0, -2, -2, ...].
a(n) = (-1)^n * A263548(n) = A128581(3*n + 2) = A190611(3*n + 2).
a(2*n) = A263571(n). a(2*n + 1) = -2 * A128582(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).
Showing 1-8 of 8 results.

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