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 1-4 of 4 results.

A164617 Expansion of (phi^3(q^3) / phi(q)) * (psi(-q^3) / psi^3(-q)) in powers of q where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 4, 10, 20, 39, 76, 140, 244, 415, 696, 1140, 1820, 2861, 4448, 6816, 10292, 15372, 22756, 33356, 48408, 69683, 99600, 141312, 199036, 278557, 387608, 536230, 737632, 1009464, 1374888, 1863764, 2514868, 3378948, 4521672, 6027000, 8002676
Offset: 0

Views

Author

Michael Somos, Aug 17 2009

Keywords

Comments

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

Examples

			G.f. = 1 + q + 4*q^2 + 10*q^3 + 20*q^4 + 39*q^5 + 76*q^6 + 140*q^7 + 244*q^8 + ...

Links

Crossrefs

Cf. A113973, A128641, A132974, A164616.

Programs

  • Mathematica
    nmax=60; CoefficientList[Series[Product[(1-x^(6*k))^14 / ((1-x^k) * (1-x^(2*k))^2 * (1-x^(3*k))^5 * (1-x^(4*k)) * (1-x^(12*k))^5),{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^6 + A)^14 / (eta(x + A) * eta(x^2 + A)^2 * eta(x^3 + A)^5 * eta(x^4 + A) * eta(x^12 + A)^5), n))};

Formula

Expansion of eta(q^6)^14 / (eta(q) * eta(q^2)^2 * eta(q^3)^5 * eta(q^4) * eta(q^12)^5) in powers of q.
Euler transform of period 12 sequence [ 1, 3, 6, 4, 1, -6, 1, 4, 6, 3, 1, 0, ...].
Convolution of A113973 and A132974. a(n) = A164616(3*n).
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(9/4) * n^(3/4)). - Vaclav Kotesovec, Oct 13 2015
A128641(n) = (-1)^n*a(n). - Michael Somos, Apr 24 2023

A164615 Expansion of c(q^2)^2 / (c(-q) * c(-q^3)) in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 1, 1, 0, 1, 2, 0, 0, 1, 0, -2, -4, 0, -2, -8, 0, 1, -2, 0, 4, 14, 0, 4, 24, 0, -1, 6, 0, -8, -38, 0, -8, -63, 0, 2, -16, 0, 14, 92, 0, 14, 150, 0, -4, 36, 0, -24, -208, 0, -23, -329, 0, 6, -78, 0, 40, 440, 0, 38, 684, 0, -10, 160, 0, -63, -884, 0, -60
Offset: 0

Views

Author

Michael Somos, Aug 17 2009

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + q + q^2 + q^4 + 2*q^5 + q^8 - 2*q^10 - 4*q^11 - 2*q^13 - 8*q^14 + ...

Links

Crossrefs

Cf. A128111, A164614, A164616, A182034. A258111.

Programs

  • Mathematica
    eta[x_] := QPochhammer[x]; A164615[n_] := SeriesCoefficient[eta[q^2]* eta[q^3]^2*eta[q^9]^3*eta[q^12]^2*eta[q^36]^3/(eta[q]*eta[q^4] *eta[q^18]^9), {q, 0, n}]; Table[A164615[n], {n,0,50}] (* G. C. Greubel, Aug 10 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A) * eta(x^3 + A)^2 * eta(x^9 + A)^3 * eta(x^12 + A)^2 * eta(x^36 + A)^3 / (eta(x + A) * eta(x^4 + A) * eta(x^18 + A)^9), n))};

Formula

Expansion of (chi(q^3) * psi(-q^3)^2)^2 / (psi(-q) * f(q^9)^2) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2) * eta(q^3)^2 * eta(q^9)^3 * eta(q^12)^2 * eta(q^36)^3 / (eta(q) * eta(q^4) * eta(q^18)^9) in powers of q.
Euler transform of period 36 sequence [ 1, 0, -1, 1, 1, -2, 1, 1, -4, 0, 1, -3, 1, 0, -1, 1, 1, 4, 1, 1, -1, 0, 1, -3, 1, 0, -4, 1, 1, -2, 1, 1, -1, 0, 1, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = (1/3) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A258111. - Michael Somos, May 20 2015
a(3*n) = 0 unless n=0. a(3*n + 1) = A128111(n). a(3*n + 2) = A164614(n).
Convolution inverse of A164616.
a(n) = (-1)^n * A182034(n). - Michael Somos, May 20 2015

A258100 Expansion of c(q) * c(q^3) / c(q^2)^2 in powers of q where c() is a cubic AGM theta function.

Original entry on oeis.org

1, 1, 0, -1, -2, 0, 4, 5, 0, -10, -12, 0, 20, 26, 0, -39, -50, 0, 76, 92, 0, -140, -168, 0, 244, 295, 0, -415, -496, 0, 696, 818, 0, -1140, -1332, 0, 1820, 2126, 0, -2861, -3324, 0, 4448, 5126, 0, -6816, -7824, 0, 10292, 11793, 0, -15372, -17548, 0, 22756
Offset: 0

Views

Author

Michael Somos, May 20 2015

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + q - q^3 - 2*q^4 + 4*q^6 + 5*q^7 - 10*q^9 - 12*q^10 + 20*q^12 + ...

Links

Crossrefs

Cf. A128641, A164616, A182034, A258099.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] QPochhammer[ q^3]^2 / (2 q^(1/8) QPochhammer[ q^6]^6), {q, 0, n}];
a[ n_] := SeriesCoefficient[ 4 q QPochhammer[ q^9]^3 EllipticTheta[ 2, 0, q^(1/2)] / (QPochhammer[ q^3] EllipticTheta[ 2, 0, q^(3/2)]^3), {q, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^3 + A)^2 * eta(x^9 + A)^3 / (eta(x + A) * eta(x^6 + A)^6), n))};

Formula

Expansion of (psi(q) * f(-q^9)^3) / (chi(-q^3)^2 * psi(q^3)^4) in powers of q where psi(), chi(), f() are Ramanujan theta functions.
Expansion of eta(q^2)^2 * eta(q^3)^2 * eta(q^9)^3 / (eta(q) * eta(q^6)^6) in powers of q.
Euler transform of period 18 sequence [ 1, -1, -1, -1, 1, 3, 1, -1, -4, -1, 1, 3, 1, -1, -1, -1, 1, 0, ...].
a(n) = (-1)^n * A164616(n). a(3*n) = A128641(n). a(3*n + 1) = A258099(n). a(3*n + 2) = 0.
Convolution invserse is A182034.

A258108 Expansion of b(-q) * b(q^6) / (b(q^3) * b(q^12)) in powers of q where b() is a cubic AGM theta function.

Original entry on oeis.org

1, 3, 0, -3, 6, 0, -12, 15, 0, -30, 36, 0, -60, 78, 0, -117, 150, 0, -228, 276, 0, -420, 504, 0, -732, 885, 0, -1245, 1488, 0, -2088, 2454, 0, -3420, 3996, 0, -5460, 6378, 0, -8583, 9972, 0, -13344, 15378, 0, -20448, 23472, 0, -30876, 35379, 0, -46116, 52644
Offset: 0

Views

Author

Michael Somos, May 20 2015

Keywords

Comments

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

Examples

			G.f. = 1 + 3*q - 3*q^3 + 6*q^4 - 12*q^6 + 15*q^7 - 30*q^9 + 36*q^10 + ...

Links

Crossrefs

Cf. A132977, A164616, A164617.

Programs

  • Mathematica
    QP=QPochhammer; A258108[n_] :=SeriesCoefficient[(QP[x^2]^9*QP[x^9]*QP[x^36])/(QP[x]^3*QP[x^3]^2*QP[x^4]^3*QP[x^12]^2*QP[x^18]), {x, 0, n}]; Table[A258108[n], {n, 0, 50}] (* G. C. Greubel, Oct 18 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^9 * eta(x^9 + A) * eta(x^36 + A) / (eta(x + A)^3 * eta(x^3 + A)^2 * eta(x^4 + A)^3 * eta(x^12 + A)^2 * eta(x^18 + A)), n))};

Formula

Expansion of eta(q^2)^9 * eta(q^9) * eta(q^36) / (eta(q)^3 * eta(q^3)^2 * eta(q^4)^3 * eta(q^12)^2 * eta(q^18)) in powers of q.
Euler transform of period 36 sequence [ 3, -6, 5, -3, 3, -4, 3, -3, 4, -6, 3, 1, 3, -6, 5, -3, 3, -4, 3, -3, 5, -6, 3, 1, 3, -6, 4, -3, 3, -4, 3, -3, 5, -6, 3, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (36 t)) = 3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A164616.
a(3*n + 2) = 0. a(3*n + 1) = 3 * A132977(n). a(3*n) = -3 * A164617(n) unless n = 0.
Showing 1-4 of 4 results.

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