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.

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

A187427 Expansion of q^(3/8) * eta(q)^3 / eta(q^3)^4 in powers of q.

Original entry on oeis.org

1, -3, 0, 9, -12, 0, 27, -42, 0, 82, -111, 0, 207, -279, 0, 486, -630, 0, 1055, -1362, 0, 2205, -2775, 0, 4374, -5472, 0, 8427, -10389, 0, 15696, -19224, 0, 28539, -34614, 0, 50630, -61059, 0, 88119, -105483, 0, 150417, -179178, 0, 252727, -299325, 0, 418068
Offset: 0

Views

Author

Michael Somos, Mar 09 2011

Keywords

Examples

			1 - 3*x + 9*x^3 - 12*x^4 + 27*x^6 - 42*x^7 + 82*x^9 - 111*x^10 + ...
q^-3 - 3*q^5 + 9*q^21 - 12*q^29 + 27*q^45 - 42*q^53 + 82*q^69 - 111*q^77 + ...

Links

Crossrefs

Cf. A053762, A187248.

Programs

  • Mathematica
    QP = QPochhammer; s = QP[q]^3/QP[q^3]^4 + O[q]^50; CoefficientList[s, q] (* Jean-François Alcover, Nov 25 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; CoefficientList[Series[q^(3/8) *eta[q]^3/ eta[q^3]^4, {q, 0, 50}], q] (* G. C. Greubel, Aug 14 2018 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^3 / eta(x^3 + A)^4, n))}

Formula

Euler transform of period 3 sequence [ -3, -3, 1, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (1728 t)) = (9/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A187428.
G.f.: Product_{k>0} (1 - x^k)^3 / (1 - x^(3*k))^4.
a(3*n) = A053762(n). a(3*n + 1) = -3 * A187428(n). a(3*n + 2) = 0.

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.

A187429 Expansion of q^(3/8) * a(q) / eta(q^3)^3 in powers of q where a() is a cubic AGM function.

Original entry on oeis.org

1, 6, 0, 9, 24, 0, 27, 84, 0, 82, 222, 0, 207, 558, 0, 486, 1260, 0, 1055, 2724, 0, 2205, 5550, 0, 4374, 10944, 0, 8427, 20778, 0, 15696, 38448, 0, 28539, 69228, 0, 50630, 122118, 0, 88119, 210966, 0, 150417, 358356, 0, 252727, 598650, 0, 418068, 986022
Offset: 0

Views

Author

Michael Somos, Mar 09 2011

Keywords

Examples

			G.f. = 1 + 6*x + 9*x^3 + 24*x^4 + 27*x^6 + 84*x^7 + 82*x^9 + 222*x^10 + ...
G.f. = q^-3 + 6*q^5 + 9*q^21 + 24*q^29 + 27*q^45 + 84*q^53 + 82*q^69 + 222*q^77 + ...

Links

Crossrefs

Cf. A053762, A187428.

Programs

  • Mathematica
    a[n_] := Module[{A = x*O[x]^n}, SeriesCoefficient[(QPochhammer[x + A]^3 + 9*x*QPochhammer[x^9 + A]^3)/QPochhammer[x^3 + A]^4, {x, 0, n}]]; Table[a[n], {n, 0, 50}] (* Jean-François Alcover, Nov 06 2015, adapted from PARI *)
  • 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^3 + A)^4, n))}

Formula

Expansion of q^(3/8) * (eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3)^4 in powers of q.
G.f. is a period 1 Fourier series which satisfies f(-1 / (192 t)) = (3/8)^(1/2) (t/i)^(-1/2) g(t) where q = exp(2 Pi i t) and g() is g.f. for A053762.
a(3*n) = A053762(n). a(3*n+ 1) = 6 * A187428(n). a(3*n + 2) = 0.
Showing 1-5 of 5 results.

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

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