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 1-6 of 6 results.

A128582 Expansion of f(x^4, x^12) * f(x, x^5) in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 0, 0, 1, 1, 0, 0, 1, 2, 0, 0, 0, 1, 0, 0, 3, 1, 0, 0, 1, 3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 2, 2, 0, 0, 1, 2, 0, 0, 2, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0, 0, 2, 1, 0, 0, 1, 2, 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 + x^4 + 2*x^5 + x^8 + x^9 + 2*x^12 + x^13 + x^16 + x^17 + 2*x^20 + ...
G.f. = q^5 + q^11 + q^29 + 2*q^35 + q^53 + q^59 + 2*q^77 + q^83 + q^101 + q^107 + ...

Links

Crossrefs

Cf. A128580, A128583, A128591.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[{m = 6 n + 5}, -1/2 DivisorSum[ m, KroneckerSymbol[ -12, #] KroneckerSymbol[ 2, m/#] &]]]; (* Michael Somos, Nov 15 2015 *)
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-7/8) EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 2, Pi/4, x]^2 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Nov 15 2015 *)
  • PARI
    {a(n) = if( n<0, 0, n = 6*n + 5; -1/2 * 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)^2 * eta(x^3 + A) * eta(x^8 + A)^2 * eta(x^12 + A) / (eta(x + A) * eta(x^4 + A)^2 * eta(x^6 + A)), n))};

Formula

Expansion of psi(x) * psi(-x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q^2)^2 * eta(q^3) * eta(q^8)^2 * eta(q^12) / (eta(q) * eta(q^4)^2 * eta(q^6)) in powers of q.
Euler transform of period 24 sequence [ 1, -1, 0, 1, 1, -1, 1, -1, 0, -1, 1, 0, 1, -1, 0, -1, 1, -1, 1, 1, 0, -1, 1, -2, ...].
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k)) * (1 + x^(4*k))^2 * (1 - x^(6*k - 3)) * (1 - x^(12*k)).
A128580(3*n + 2) = -2 * a(n). a(4*n) = A128583. a(4*n + 1) = A128591(n). a(4*n + 2) = a(4*n + 3) = 0.

A190615 Expansion of f(x^2) * f(x^3) / (chi(x) * chi(x^6)) in powers of x where f(), chi() 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, May 14 2011

Keywords

Comments

Number 63 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 + ...

Links

Crossrefs

Cf. A113700, A128580, A128583, A128591, A129402, A134177, A257920, A259895, A259896, A260089, A260110, A260118, A260308.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, (-1)^n DivisorSum[ 2 n + 1, KroneckerSymbol[ -6, #] &]]; (* Michael Somos, Jun 09 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 09 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x^2] QPochhammer[ -x^3] / (QPochhammer[ -x, x^2] QPochhammer[ -x^6, x^12]), {x, 0, n}]; (* Michael Somos, Jun 09 2015 *)
  • PARI
    {a(n) = if( n<0, 0, (-1)^n * sumdiv( 2*n + 1, d, kronecker( -6, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^4 + A)^4 * eta(x^6 + A)^4 * eta(x^24 + A) / (eta(x^2 + A)^3 * eta(x^3 + A) * eta(x^8 + A) * eta(x^12 + A)^3), n))};
  • PARI
    {a(n) = my(A, p, e); if( n<0, 0, A = factor(2*n + 1); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if( p==2, 0, p==3, (-1)^e, p%24 < 12, (e+1) * if( p%24 < 6, 1, (-1)^e), (1 + (-1)^e) / 2 )))};

Formula

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.
Expansion of q^(-1/2) * eta(q) * eta(q^4)^4 * eta(q^6)^4 * eta(q^24) / (eta(q^2)^3 * eta(q^3) * eta(q^8) * eta(q^12)^3) in powers of q.
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, 5 (mod 24), b(p^e) = (-1)^e * (e+1) if p == 7, 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, -2, -1, -1, -1, -1, 0, 2, -1, -2, -1, 2, 0, -1, -1, -1, -1, -2, 0, 2, -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).
G.f.: Sum_{k>0} Kronecker( 6, k) * q^k / (1 + q^(2*k)) = Sum_{k>=0} a(k) * q^(2*k + 1).
G.f.: Product_{k>0} (1 + (-x)^k) * (1 - (-x^2)^k) * (1 - (-x^3)^k) * (1 + (-x^6)^k).
a(n) = (-1)^n * A129402(n). a(3*n + 1) = -a(n). a(12*n + 6) = a(12*n + 8) = a(12*n + 9) = a(12*n + 11) = 0.
a(12*n) = A113700(n). a(12*n + 2) = 2 * A128583(n). a(12*n + 5) = -2 * A128591(n). - Michael Somos, Jun 09 2015
a(n) = (-1)^floor(n/2) * A128580(n) = (-1)^(n + floor(n/2)) * A134177(n). - Michael Somos, Jul 29 2015
a(3*n) = A260110(n). a(3*n + 2) = 2 * A260118(n). - Michael Somos, Jul 29 2015
a(4*n) = A260308(n). a(4*n + 1) = - A257920(n). a(4*n + 2) = 2 * A259895(n). a(4*n + 3) = -2 * A259896(n). - Michael Somos, Jul 29 2015
a(12*n + 3) = -2 * A260089(n). - Michael Somos, Jul 29 2015

A259895 Expansion of psi(x^2) * psi(x^3) in powers of x where psi() is a Ramanujan theta function.

Original entry on oeis.org

1, 0, 1, 1, 0, 1, 1, 0, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 1, 0, 2, 1, 0, 1, 1, 0, 0, 0, 0, 1, 3, 0, 1, 1, 0, 0, 1, 0, 1, 1, 0, 0, 2, 0, 0, 2, 0, 1, 1, 0, 1, 2, 0, 0, 0, 0, 1, 1, 0, 1, 2, 0, 0, 1, 0, 3, 0, 0, 0, 1, 0, 0, 2, 0, 1, 3, 0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 2
Offset: 0

Views

Author

Michael Somos, Jul 07 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^3 + x^5 + x^6 + 2*x^9 + x^11 + x^12 + 2*x^15 + x^18 + ...
G.f. = q^5 + q^21 + q^29 + q^45 + q^53 + 2*q^77 + q^93 + q^101 + 2*q^125 + ...

Links

Crossrefs

Cf. A000377, A128580, A128583, A129402, A134177, A190615, A192013, A257921, A259668, A259896.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] EllipticTheta[ 2, 0, x^(3/2)] / (4 q^(5/8)), {x, 0, n}];
a[ n_] := If[ n < 0, 0, 1/2 Sum[ KroneckerSymbol[ -6, d], {d, Divisors[8 n + 5]}]]; (* Michael Somos, Jul 22 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 * eta(x^6 + A)^2 / (eta(x^2 + A) * eta(x^3 + A)), n))};
  • PARI
    {a(n) = if( n<0, 0, 1/2 * sumdiv( 8*n + 5, d, kronecker( -6, d)))};

Formula

Expansion of q^(-5/8) * eta(q^4)^2 * eta(q^6)^2 / (eta(q^2) * eta(q^3)) in powers of q.
Euler transform of period 12 sequence [ 0, 1, 1, -1, 0, 0, 0, -1, 1, 1, 0, -2, ...].
a(n) = A259896(3*n + 1). a(3*n) = A128583(n). a(3*n + 1) = a(9*n + 8) = 0.
2 * a(n) = A129402(4*n + 2) = A190615(4*n + 2) = A000377(8*n + 5) = A192013(8*n + 5). - Michael Somos, Jul 22 2015
-2 * a(n) = A259668(2*n + 1) = A128580(4*n + 2) = A134177(4*n + 2) = A257921(6*n + 3). - Michael Somos, Jul 22 2015
a(3*n + 2) = A259896(n). - Michael Somos, Jul 22 2015

A260118 Expansion of f(-x, -x^5) * psi(x^4) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -1, 0, 0, 1, -2, 0, 0, 1, -1, 0, 0, 2, -1, 0, 0, 1, -1, 0, 0, 2, -1, 0, 0, 1, -2, 0, 0, 1, -1, 0, 0, 1, -2, 0, 0, 0, -1, 0, 0, 3, -1, 0, 0, 1, -3, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 2, -1, 0, 0, 2, -2, 0, 0, 1, -2, 0, 0, 2, -1, 0, 0, 0, -1, 0, 0, 1, -1, 0, 0, 2
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 - x + x^4 - 2*x^5 + x^8 - x^9 + 2*x^12 - x^13 + x^16 - x^17 + ...
G.f. = q^5 - q^11 + q^29 - 2*q^35 + q^53 - q^59 + 2*q^77 - q^83 + q^101 + ...

Links

Crossrefs

Cf. A128538, A128582, A128591, A134177, A190615.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x]^2 EllipticTheta[ 2, 0, x^(3/2)]/(4 x^(7/8) QPochhammer[ -x]), {x, 0, n}];
a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, (-1)^n DivisorSum[ m, {-1, 0, 0, 0, 1, 0}[[Mod[#, 6, 1]]] {1, 0, 0, 0, 0, 0, 1, 0}[[Mod[m/#, 8, 1]]] &]]];
  • PARI
    {a(n) = my(m); if( n<0, 0, m=6*n + 5; (-1)^n * sumdiv(m, d, ((d%6==5) - (d%6==1)) * ((m/d%8==1) + (m/d%8==7))))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^6 + A)^2 * eta(x^8 + A)^2 / (eta(x^2 + A) * eta(x^3 + A) * eta(x^4 + A)), n))};

Formula

Expansion of psi(-x^2)^2 * psi(x^3) / f(x) in powers of x where psi(), f() are Ramanujan theta functions.
Expansion of psi(-x) * psi(x^3) / chi(-x^4)^2 in powers of x where psi(), chi() are Ramanujan theta functions.
Expansion of q^(-5/6) * eta(q) * eta(q^6)^2 * eta(q^8)^2 / (eta(q^2) * eta(q^3) * eta(q^4)) in powers of q.
Euler transform of period 24 sequence [ -1, 0, 0, 1, -1, -1, -1, -1, 0, 0, -1, 0, -1, 0, 0, -1, -1, -1, -1, 1, 0, 0, -1, -2, ...].
a(n) = (-1)^n * A128582(n). 2 * a(n) = - A134177(3*n + 2) = A190615(3*n + 2).
a(4*n) = A128583(n). a(4*n + 1) = - A128591(n). a(4*n + 2) = a(4*n + 3) = 0.

A229723 Expansion of psi(q) * chi(-q^3) * phi(-q^6) in powers of q where phi(), psi(), chi() are Ramanujan theta functions.

Original entry on oeis.org

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

Views

Author

Michael Somos, Sep 27 2013

Keywords

Comments

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

Examples

			G.f. = 1 + q - q^4 - 2*q^6 - 2*q^7 - 2*q^9 + 2*q^10 + 4*q^15 - q^16 + ...

Links

Crossrefs

Cf. A128583.

Programs

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

Formula

Expansion eta(q^2)^2 * eta(q^3) * eta(q^6) / (eta(q) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 1, -1, 0, -1, 1, -3, 1, -1, 0, -1, 1, -2, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (288 t)) = 13824^(1/2) (t / i) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A128583.
a(3*n + 2) = 0.

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 1-6 of 6 results.

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

Content is available under The OEIS End-User License Agreement.