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.

A053762 Number of 3-colored generalized Frobenius partitions of n.

Original entry on oeis.org

1, 9, 27, 82, 207, 486, 1055, 2205, 4374, 8427, 15696, 28539, 50630, 88119, 150417, 252727, 418068, 682344, 1099343, 1750968, 2758185, 4301682, 6645150, 10175625, 15451744, 23281686, 34819227, 51712860, 76292784, 111850740, 162997314
Offset: 0

Views

Author

James Sellers, Apr 05 2000

Keywords

Comments

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

Examples

			1 + 9*x + 27*x^2 + 82*x^3 + 207*x^4 + 486*x^5 + 1055*x^6 + 2205*x^7 + ...
1/q + 9*q^7 + 27*q^15 + 82*q^23 + 207*q^31 + 486*q^39 + 1055*q^47 + 2205*q^55 + ...

Links

Crossrefs

Cf. A000716, A004016, A051136.

Programs

  • Mathematica
    nmax = 30; CoefficientList[Series[(Product[(1 - x^k)^3, {k, 1, nmax}] + 9*x*Product[(1 - x^(9*k))^3, {k, 1, nmax}]) / Product[((1 - x^k)^3*(1 - x^(3*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 13 2016 *)
a[n_]:= SeriesCoefficient[q^(1/8)*(eta[q]^3 + 9*eta[q^9]^3)/(eta[q]^3* eta[q^3]), {q, 0, n}]; Table[a[n], {n,0,50}] (* G. C. Greubel, Feb 08 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A)^3 + 9 * x * eta(x^9 + A)^3) / (eta(x + A)^3 * eta(x^3 + A)), n))} /* Michael Somos, Mar 09 2011 */

Formula

Expansion of q^(1/8) * (eta(q)^3 + 9 * eta(q^9)^3) / (eta(q)^3 * eta(q^3)) in powers of q. - Michael Somos, Mar 09 2011
Expansion of a(x) / f(-x)^3 in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Aug 21 2012
Convolution of A000716 and A004016. - Michael Somos, Mar 09 2011
a(n) ~ exp(sqrt(2*n)*Pi)/(4*sqrt(3)*n). - Vaclav Kotesovec, Nov 13 2016

A137828 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, 12072, 0, 0, 16286
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, A137829.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 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^2 + A)^5 / eta(x^4 + A)^4 / eta(x + A)^2, n))};

Formula

Expansion of q^(1/3) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^4) 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 / (144 t)) = 6^(-1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A051136.
G.f.: Product_{k>0} (1 + x^k) / ( (1 - x^k) * (1 + x^(2*k))^4 ).
a(4*n + 2) = a(4*n + 3) = 0.
a(4*n) = A051136(n). a(4*n + 1) = 2 * A137829(n).

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).

A247663 Number of 4-colored generalized Frobenius partitions.

Original entry on oeis.org

1, 16, 68, 256, 777, 2160, 5460, 13056, 29482, 63952, 133456, 270080, 531091, 1019424, 1913156, 3520512, 6360765, 11305488, 19789160, 34159616, 58201535
Offset: 0

Views

Author

N. J. A. Sloane, Oct 05 2014

Keywords

References

  • G. E. Andrews, "Generalized Frobenius Partitions," AMS Memoir 301, 1984 (sequence is denoted c\phi_4(n)).

Crossrefs

Cf. A000041, A051136, A053762, A247664.

A247664 Number of 5-colored generalized Frobenius partitions.

Original entry on oeis.org

1, 25, 150, 675, 2450, 7876, 22825, 61550, 155925, 375875, 867627, 1930775, 4159100, 8708150, 17771725, 35447503, 69243250, 132718300, 249979625, 463348600, 846149380
Offset: 0

Views

Author

N. J. A. Sloane, Oct 05 2014

Keywords

References

  • G. E. Andrews, "Generalized Frobenius Partitions," AMS Memoir 301, 1984 (sequence is denoted c\phi_5(n)).

Crossrefs

Cf. A000041, A051136, A053762, A247663.
Showing 1-5 of 5 results.

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

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