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 1-3 of 3 results.

A051136 Number of 2-colored generalized Frobenius partitions.

Original entry on oeis.org

1, 4, 9, 20, 42, 80, 147, 260, 445, 744, 1215, 1944, 3059, 4740, 7239, 10920, 16286, 24028, 35110, 50844, 73010, 104028, 147144, 206700, 288501, 400232, 552037, 757288, 1033495, 1403508, 1897088, 2552812, 3420527, 4564500, 6067265
Offset: 0

Views

Author

James Sellers

Keywords

Comments

Ramanujan theta functions: f(q) := Product_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Product_{k>=0} (1+q^(2k+1)) (A000700).

Examples

			1 + 4*x + 9*x^2 + 20*x^3 + 42*x^4 + 80*x^5 + 147*x^6 + 260*x^7 + ...
1/q + 4*q^11 + 9*q^23 + 20*q^35 + 42*q^47 + 80*q^59 + 147*q^71 + ...

References

  • G. E. Andrews, "Generalized Frobenius Partitions," AMS Memoir 301, 1984 (sequence is denoted c\phi_2(n)).
  • G. E. Andrews, q-series, CBMS Regional Conference Series in Mathematics, 66, Amer. Math. Soc. 1986, see p. 67, Eq. (7.20). MR0858826 (88b:11063)

Links

Crossrefs

Cf. A000041, A053762, A247663, A247664.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[(1 + x^(2*k-1)) / ((1 - x^(2*k-1))^3 * (1 - x^(4*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 31 2015 *)
QP = QPochhammer; s = QP[q^2]^5 / QP[q]^4 / QP[q^4]^2 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Nov 09 2015, adapted from PARI *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / eta(x + A)^4 / eta(x^4 + A)^2, n))} /* Michael Somos, Feb 12 2008 */

Formula

Expansion of phi(q) / f(-q)^2 in powers of q where phi(), f() are Ramanujan theta functions.
Expansion of q^(1/12) * eta(q^2)^5 / (eta(q)^4 * eta(q^4)^2) in powers of q. - Michael Somos, Apr 25 2003
Euler transform of period 4 sequence [4, -1, 4, 1, ...]. - Michael Somos, Apr 25 2003
G.f. is a period 1 Fourier series which satisfies f(-1 / (144 t)) = 24^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is g.f. for A137828.
G.f.: Product_{k>0} (1 -x^(4*k-2)) / ((1 - x^(2*k-1))^4 * (1 - x^(4*k))). [Andrews, Memoir, p. 13, equation (5.17)]
G.f.: Product_{k>0} (1 + x^k)^3 / ((1 - x^k) * (1 + x^(2*k))^2). - Michael Somos, Feb 12 2008
a(n) ~ exp(2*Pi*sqrt(n/3)) / (4*sqrt(3)*n). - Vaclav Kotesovec, Aug 31 2015

A137829 Expansion of psi(q^2) / f(-q)^2 in powers of q where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 6, 12, 25, 46, 86, 148, 255, 420, 686, 1088, 1712, 2634, 4020, 6036, 8988, 13214, 19282, 27840, 39923, 56750, 80160, 112384, 156660, 216958, 298894, 409420, 558119, 756950, 1022090, 1373760, 1838932, 2451366, 3255480, 4306920, 5678104, 7459634, 9768386
Offset: 0

Views

Author

Michael Somos, Feb 12 2008

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Moreno (preprint) calls this |Phi_n|. - N. J. A. Sloane, Sep 01 2018

Examples

			G.f. = 1 + 2*x + 6*x^2 + 12*x^3 + 25*x^4 + 46*x^5 + 86*x^6 + 148*x^7 + ...
G.f. = q + 2*q^7 + 6*q^13 + 12*q^19 + 25*q^25 + 46*q^31 + 86*q^37 + 148*q^43 + ...

References

  • Lusztig, G., Irreducible representation of finite classical groups, Inventiones Math., 43 (1977), 125-175. See p. 135.
  • Moreno, Carlos J., Partitions, congruences and Kac-Moody Lie algebras. Preprint, 37pp., no date. See Table II.

Links

Crossrefs

Cf. A137828, A137830.
Cf. Half of A201078, which gives another application.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]^2), {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
nmax=60; CoefficientList[Series[Product[(1+x^k) * (1+x^(2*k))^2 / (1-x^k),{k,1,nmax}],{x,0,nmax}],x] (* Vaclav Kotesovec, Oct 13 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A)^2 * eta(x^2 + A)), n))};

Formula

Expansion of q^(-1/6) * eta(q^4)^2 / (eta(q)^2 * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [ 2, 3, 2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 96^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137830.
G.f.: Product_{k>0} (1 + x^k) * (1 + x^(2*k))^2 / (1 - x^k).
2 * a(n) = A137828(4*n + 1).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (8*sqrt(3)*n). - Vaclav Kotesovec, Oct 13 2015

A137830 Expansion of phi(-x) / f(-x^4)^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 0, 0, 4, -4, 0, 0, 9, -12, 0, 0, 20, -24, 0, 0, 42, -50, 0, 0, 80, -92, 0, 0, 147, -172, 0, 0, 260, -296, 0, 0, 445, -510, 0, 0, 744, -840, 0, 0, 1215, -1372, 0, 0, 1944, -2176, 0, 0, 3059, -3424, 0, 0, 4740, -5268, 0, 0, 7239, -8040, 0, 0, 10920
Offset: 0

Views

Author

Michael Somos, Feb 12 2008

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^4 - 4*x^5 + 9*x^8 - 12*x^9 + 20*x^12 - 24*x^13 + 42*x^16 + ...
G.f. = 1/q - 2*q^2 + 4*q^11 - 4*q^14 + 9*q^23 - 12*q^26 + 20*q^35 - 24*q^38 + ...

Links

Crossrefs

Cf. A051136, A137828, A137829.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, x] / QPochhammer[ x^4]^2, {x, 0, n}]; (* Michael Somos, Oct 04 2015 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^2 / eta(x^2 + A) / eta(x^4 + A)^2, n))};

Formula

Expansion of q^(1/3) * eta(q)^2 / (eta(q^2) * eta(q^4)^2) in powers of q.
Euler transform of period 4 sequence [ -2, -1, -2, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = (4/3)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A137829.
G.f.: ( Product_{k>0} (1 - x^(2*k)) * (1 + x^k)^2 * (1 + x^(2*k))^2 )^(-1).
a(4*n + 2) = a(4*n + 3) = 0.
a(n) = (-1)^n * A137828(n). a(4*n) = A051136(n). a(4*n + 1) = -2 * A137829(n).
Showing 1-3 of 3 results.

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