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 1-5 of 5 results.

A260158 Expansion of psi(x)^4 * psi(-x^3) / f(x) in powers of x where psi, f() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 4, 6, 7, 6, 10, 12, 13, 15, 14, 18, 18, 21, 22, 18, 25, 27, 28, 24, 26, 33, 34, 42, 37, 30, 36, 42, 43, 45, 38, 48, 49, 42, 54, 42, 56, 57, 58, 60, 43, 63, 64, 66, 67, 63, 70, 60, 73, 84, 62, 78, 79, 72, 72, 66, 90, 87, 88, 90, 74, 78, 98, 96, 97, 78
Offset: 0

Views

Author

Michael Somos, Nov 09 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 3*x + 4*x^2 + 6*x^3 + 7*x^4 + 6*x^5 + 10*x^6 + 12*x^7 + 13*x^8 + ...
G.f. = q^7 + 3*q^23 + 4*q^39 + 6*q^55 + 7*q^71 + 6*q^87 + 10*q^103 + 12*q^119 + ...

Links

Crossrefs

Cf. A124815, A260109, A260295.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 5}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #]&] / 4]];
a[ n_] := SeriesCoefficient[ 2^(-9/2) x^(-7/8) EllipticTheta[ 2, 0, x^(1/2)]^4 EllipticTheta[ 2, Pi/4, x^(3/2)] / QPochhammer[ -x], {x, 0, n}];
  • PARI
    {a(n) = my(m); if( n<0, 0, m = 6*n + 5; sumdiv( m, d, m/d * kronecker( 12, d)) / 4)};
  • PARI
    {a(n) = if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x + A)^3 * eta(x^6 + A)), n))};

Formula

Expansion of q^(-5/6) * eta(q^2)^5 * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q)^3 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 3, -2, 2, -3, 3, -2, 3, -3, 2, -2, 3, -4, ...].
4 * a(n) = A260109(3*n + 2) = A124815(6*n + 5).
a(2*n + 1) = 3 * A260295(n).

A109039 Expansion of eta(q) * eta(q^3) * (eta(q^4) * eta(q^6) / eta(q^12))^2 in powers of q.

Original entry on oeis.org

1, -1, -1, -1, -1, 4, -1, 6, -1, -1, 4, -12, -1, -14, 6, 4, -1, 16, -1, 18, 4, 6, -12, -24, -1, -21, -14, -1, 6, 28, 4, 30, -1, -12, 16, -24, -1, -38, 18, -14, 4, 40, 6, 42, -12, 4, -24, -48, -1, -43, -21, 16, -14, 52, -1, 48, 6, 18, 28, -60, 4, -62, 30, 6
Offset: 0

Views

Author

Michael Somos, Jun 17 2005

Keywords

Comments

Number 25 of the 74 eta-quotients listed in Table I of Martin (1996).

Examples

			G.f. = 1 - q - q^2 - q^3 - q^4 + 4*q^5 - q^6 + 6*q^7 - q^8 - q^9 + 4*q^10 + ...

Links

Crossrefs

Cf. A109040, A124815.

Programs

  • Magma
    A := Basis( ModularForms( Gamma1(12), 2), 64); A[1] - A[2] - A[3] - A[4] - A[5] + 4*A[6] - A[7] + 6*A[8] - A[9]; /* Michael Somos, May 18 2015 */
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q] QPochhammer[ q^3] (QPochhammer[ q^4] QPochhammer[ q^6] / QPochhammer[ q^12])^2, {q, 0, n}]; (* Michael Somos, May 18 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q^3] QPochhammer[ q^3, q^6]^3 EllipticTheta[ 2, 0, q^(1/2)] EllipticTheta[ 2, Pi/4, q^(1/2)]^2 / (4 q^(3/8)), {q, 0, n}]; (* Michael Somos, May 18 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A) * eta(x^3 + A) * eta(x^4 + A)^2 * eta(x^6 + A)^2 / eta(x^12 + A)^2, n))};

Formula

Euler transform of period 12 sequence [ -1, -1, -2, -3, -1, -4, -1, -3, -2, -1, -1, -4, ...].
G.f.: Product_{k>0} (1 - x^k) * (1 - x^(3*k)) * (1 - x^(4*k))^2 / (1 + x^(6*k))^2.
a(n) = -A109040(n) unless n=0. a(2*n) = a(3*n) = a(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A124815. - Michael Somos, May 18 2015
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(3)) = 0.237425... . - Amiram Eldar, Jan 29 2024

A260109 Expansion of f(x^3) * f(-x^3)^2 * psi(x)^2 / psi(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 3, 4, 6, 9, 12, 14, 12, 16, 18, 18, 24, 21, 27, 28, 30, 36, 24, 38, 42, 40, 42, 36, 48, 43, 48, 52, 48, 54, 60, 62, 54, 56, 66, 72, 72, 74, 63, 72, 78, 81, 84, 64, 84, 88, 84, 90, 72, 98, 108, 100, 102, 72, 108, 110, 114, 112, 96, 126, 96, 133, 120, 104
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 + 3*x + 4*x^2 + 6*x^3 + 9*x^4 + 12*x^5 + 14*x^6 + 12*x^7 + ...
G.f. = q + 3*q^3 + 4*q^5 + 6*q^7 + 9*q^9 + 12*q^11 + 14*q^13 + 12*q^15 + ...

Links

Crossrefs

Cf. A124815.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ 1/4 x^(-1/2) EllipticTheta[ 2, 0, x^(1/2)]^2 EllipticTheta[ 2, Pi/4, x^(3/2)] EllipticTheta[ 4, 0, x^6]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {x, 0, n}];
a[ n_] := SeriesCoefficient[ 2^(-3/2) x^(-1/8) QPochhammer[ -x^3] QPochhammer[ x^3]^2 EllipticTheta[ 2, 0, x^(1/2)]^2 / EllipticTheta[ 2, Pi/4, x^(1/2)], {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^3 + A) * eta(x^6 + A)^3 / (eta(x + A)^3 * eta(x^4 + A) * eta(x^12 + A)), n))};

Formula

Expansion of psi(-x^3) * phi(-x^6)^2 * psi(x)^2 / psi(-x) in powers of x where phi(), psi() are Ramanujan theta functions.
Expansion of phi(x) * f(x, x^5) * f(x^2, x^4)^2 in powers of x. - Michael Somos, Jul 18 2015
Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^3) * eta(q^6)^3 / (eta(q)^3 * eta(q^4) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 3, -2, 2, -1, 3, -6, 3, -1, 2, -2, 3, -4, ...].
a(n) = A124815(2*n + 1). a(3*n + 1) = 3 * a(n).

A261445 Expansion of f(x, x^3) * f(x, x^2)^3 in powers of x where f(, ) is Ramanujan's general theta function.

Original entry on oeis.org

1, 4, 9, 14, 16, 18, 21, 28, 36, 38, 40, 36, 43, 52, 54, 62, 56, 72, 74, 72, 81, 64, 88, 90, 98, 100, 72, 110, 112, 126, 133, 104, 126, 108, 136, 144, 112, 148, 144, 158, 144, 144, 183, 172, 180, 182, 152, 162, 194, 196, 198, 160, 216, 216, 180, 224, 189, 230
Offset: 0

Views

Author

Michael Somos, Aug 18 2015

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 9*x^2 + 14*x^3 + 16*x^4 + 18*x^5 + 21*x^6 + 28*x^7 + ...
G.f. = q + 4*q^5 + 9*q^9 + 14*q^13 + 16*q^17 + 18*q^21 + 21*q^25 + 28*q^29 + ...

Links

Crossrefs

Cf. A124815, A209613, A260109 A260165, A260301, A263021.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (QPochhammer[ x^3] / (QPochhammer[ x, x^6] QPochhammer[ x^5, x^6]))^3 EllipticTheta[ 2, 0, x^(1/2)] / (2 x^(1/8)), {x, 0, n}];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x]^6 EllipticTheta[ 4, 0, x^3]^3 EllipticTheta[ 4, 0, x], {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x^2]^3 EllipticTheta[ 4, 0, x^3]^3 / EllipticTheta[ 4, 0, x]^2, {x, 0, n}]; (* Michael Somos, Nov 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^3 + A)^6 / (eta(x + A)^4 * eta(x^6 + A)^3), n))};

Formula

Expansion of f(-x^2)^3 * phi(-x^3)^3 / phi(-x)^2 in powers of x where phi(), f() are Ramanujan theta functions.
Expansion of q^(-1/4) * eta(q^2)^5 * eta(q^3)^6 / (eta(q)^4 * eta(q^6)^3) in powers of q.
Euler transform of period 6 sequence [4, -1, -2, -1, 4, -4, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (24 t)) = 12^(1/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A260301. - Michael Somos, Nov 13 2015
a(n) = A260109(2*n) = A263021(3*n) = A124815(4*n + 1) = A209613(4*n + 1). - Michael Somos, Nov 13 2015
a(3*n + 1) = 4 * A260165(n). a(3*n + 2) = 9 * A263021(n). - Michael Somos, Nov 13 2015

A260114 Expansion of f(x)^4 * phi(-x^3) / phi(-x) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 6, 14, 18, 21, 30, 38, 42, 43, 48, 62, 66, 74, 78, 64, 84, 98, 102, 110, 96, 133, 126, 108, 138, 112, 150, 158, 162, 183, 126, 182, 192, 194, 198, 160, 210, 180, 222, 230, 192, 242, 252, 288, 228, 208, 270, 278, 282, 273, 240, 252, 306, 314, 336, 294, 330
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 + 6*x + 14*x^2 + 18*x^3 + 21*x^4 + 30*x^5 + 38*x^6 + 42*x^7 + ...
G.f. = q + 6*q^7 + 14*q^13 + 18*q^19 + 21*q^25 + 30*q^31 + 38*q^37 + ...

Links

Crossrefs

Cf. A113421, A124815, A260518.

Programs

  • Mathematica
    a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, # KroneckerSymbol[ -3, #] KroneckerSymbol[ -4, m/#] &]]];
a[ n_] := If[ n < 0, 0, With[ {m = 6 n + 1}, DivisorSum[ m, m/# KroneckerSymbol[ 12, #] &]]];
a[ n_] := SeriesCoefficient[ QPochhammer[ -x]^4 EllipticTheta[ 4, 0, x^3] / EllipticTheta[ 4, 0, x], {x, 0, n}];
  • PARI
    {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, d * kronecker( -3, d) * kronecker( -4, m/d)))};
  • PARI
    {a(n) = my(m = 6*n + 1); if (n<0, 0, sumdiv( m, d, m/d * kronecker( 12, d)))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^13 * eta(x^3 + A)^2 / (eta(x + A)^6 * eta(x^4 + A)^4 * eta(x^6 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q^2)^13 * eta(q^3)^2 / (eta(q)^6 * eta(q^4)^4 * eta(q^6)) in powers of q.
Euler transform of period 12 sequence [ 6, -7, 4, -3, 6, -8, 6, -3, 4, -7, 6, -4, ...].
a(n) = A113421(6*n + 1) = A124815(6*n + 1).
a(2*n + 1) = 6 * A260518(n). - Michael Somos, Oct 07 2015
Showing 1-5 of 5 results.

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