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 1-2 of 2 results.

A263021 Expansion of f(-x^3)^6 / (phi(-x) * phi(-x^3)) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 4, 4, 6, 8, 9, 10, 8, 14, 14, 16, 16, 16, 20, 18, 22, 24, 21, 26, 28, 28, 28, 24, 36, 34, 36, 38, 32, 32, 40, 42, 44, 36, 46, 56, 43, 50, 40, 52, 54, 56, 54, 42, 60, 62, 64, 64, 56, 66, 56, 72, 70, 56, 74, 74, 76, 72, 64, 80, 81, 84, 84, 64, 76, 88, 88
Offset: 0

Views

Author

Michael Somos, Oct 07 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 + 4*x^2 + 4*x^3 + 6*x^4 + 8*x^5 + 9*x^6 + 10*x^7 + 8*x^8 + ...
G.f. = q^3 + 2*q^7 + 4*q^11 + 4*q^15 + 6*q^19 + 8*q^23 + 9*q^27 + 10*q^31 + ...

Links

Crossrefs

Cf. A260295, A260518, A261445.

Programs

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

Formula

Expansion of q^(-3/4) * eta(q^2) * eta(q^3)^4 * eta(q^6) / eta(q)^2 in powers of q.
Euler transform of period 6 sequence [ 2, 1, -2, 1, 2, -4, ...].
a(3*n) = A261445(n). a(3*n + 1) = 2 * A260518(n). a(3*n + 2) = 4 * A260295(n).

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 1-2 of 2 results.

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