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.

A261369 Expansion of (psi(-x^3) / f(x))^2 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, 5, -12, 24, -46, 86, -152, 262, -442, 725, -1168, 1852, -2886, 4436, -6736, 10103, -14994, 22040, -32092, 46336, -66380, 94378, -133256, 186926, -260576, 361126, -497716, 682340, -930774, 1263624, -1707672, 2297737, -3078850, 4109022, -5462924, 7236280
Offset: 0

Views

Author

Michael Somos, Aug 16 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 + 5*x^2 - 12*x^3 + 24*x^4 - 46*x^5 + 86*x^6 - 152*x^7 + ...
G.f. = q^2 - 2*q^5 + 5*q^8 - 12*q^11 + 24*q^14 - 46*q^17 + 86*q^20 + ...

Links

Crossrefs

Cf. A139135, A187153, A213265, A262930.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, Pi/4, x^(3/2)]^2 / (2 x^(3/4) QPochhammer[ -x]^2), {x, 0, n}];
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A) / (eta(x^2 + A)^3 * eta(x^6 + A)))^2, n))};

Formula

Expansion of q^(-2/3) * (eta(q) * eta(q^3) * eta(q^4) * eta(q^12) / (eta(q^2)^3 * eta(q^6)))^2 in powers of q.
Euler transform of period 12 sequence [ -2, 4, -4, 2, -2, 4, -2, 2, -4, 4, -2, 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 A262930. - Michael Somos, Nov 07 2015
a(n) = A187153(3*n + 2) = A213265(3*n + 2) = A262930(3*n + 2). - Michael Somos, Nov 07 2015
Convolution square of A139135. - Michael Somos, Nov 07 2015
a(n) ~ (-1)^n * exp(2*Pi*sqrt(n/3)) / (8*3^(5/4)*n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A263528 Expansion of (psi(x) * psi(x^3) / f(-x^3)^2)^2 in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 1, 8, 14, 6, 38, 60, 23, 140, 208, 76, 439, 626, 221, 1232, 1704, 584, 3182, 4300, 1443, 7700, 10212, 3368, 17673, 23076, 7497, 38808, 50008, 16046, 82070, 104560, 33190, 167996, 211920, 66628, 334202, 417902, 130288, 648224, 804254, 248858, 1229148
Offset: 0

Views

Author

Michael Somos, Oct 19 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 + x^2 + 8*x^3 + 14*x^4 + 6*x^5 + 38*x^6 + 60*x^7 + 23*x^8 + ...
G.f. = q + 2*q^3 + q^5 + 8*q^7 + 14*q^9 + 6*q^11 + 38*q^13 + 60*q^15 + 23*x^17 + ...

Links

Crossrefs

Cf. A262930.

Programs

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

Formula

Expansion of q^(-1/2) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q) * eta(q^3)^3))^2 in powers of q.
Euler transform of period 6 sequence [ 2, -2, 8, -2, 2, 0, ...].
-2 * a(n) = A262930(2*n + 1).

A293387 Expansion of (eta(q^2)^2/(eta(q)eta(q^3)))^2 in powers of q.

Original entry on oeis.org

1, 2, 1, 4, 6, 2, 12, 16, 5, 28, 36, 12, 60, 76, 24, 120, 150, 46, 228, 280, 86, 416, 504, 152, 732, 878, 262, 1252, 1488, 442, 2088, 2464, 725, 3408, 3996, 1168, 5460, 6364, 1852, 8600, 9972, 2886, 13344, 15400, 4436, 20424, 23472, 6736, 30876, 35346, 10103
Offset: 0

Views

Author

Seiichi Manyama, Oct 07 2017

Keywords

Links

Crossrefs

Cf. A262930, A292577.

Programs

  • Mathematica
    nmax = 100; CoefficientList[Series[Product[((1 - x^(2*k))^2/((1 - x^k)*(1 - x^(3*k))))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Oct 08 2017 *)

Formula

G.f.: Product_{k>0} ((1 - x^(2*k))^2/((1 - x^k)*(1 - x^(3*k))))^2.

A259033 Expansion of psi(x^3)^2 * f(-x^2)^4 / f(-x)^6 in powers of where psi(), f() are Ramanujan theta function.

Original entry on oeis.org

1, 6, 23, 76, 221, 584, 1443, 3368, 7497, 16046, 33190, 66628, 130288, 248858, 465387, 853836, 1539425, 2731462, 4775703, 8236856, 14027754, 23609794, 39301171, 64747876, 105638153, 170778512, 273704800, 435079524, 686237877, 1074405242, 1670333294, 2579418528
Offset: 0

Views

Author

Michael Somos, Nov 07 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 + 23*x^2 + 76*x^3 + 221*x^4 + 584*x^5 + 1443*x^6 + 3368*x^7 + ...
G.f. = q^5 + 6*q^11 + 23*q^17 + 76*q^23 + 221*q^29 + 584*q^35 + 1443*q^41 + ...

Links

Crossrefs

Cf. A213265, A261369, A262930, A263528.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (1/4) x^(-3/4) EllipticTheta[ 2, 0, x^(3/2)]^2 QPochhammer[ x^2]^4 / QPochhammer[ x]^6, {x, 0, n}];
nmax = 40; CoefficientList[Series[Product[((1 + x^k)^2 * (1 + x^(3*k))^2 * (1 - x^(3*k)) / (1 - x^k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Nov 16 2017 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 * eta(x^6 + A)^2 / (eta(x + A)^3 * eta(x^3 + A)))^2, n))};

Formula

Expansion of q^(-5/6) * (eta(q^2)^2 * eta(q^6)^2 / (eta(q)^3 * eta(q^3)))^2 in powers of q.
Euler transform of period 6 sequence [ 6, 2, 8, 2, 6, 0, ...].
a(n) = A263528(3*n + 2). -2 * a(n) = A261369(2*n + 1) = A213265(6*n + 5) = A262930(6*n + 5).
a(n) ~ exp(2*Pi*sqrt(2*n/3)) / (2^(19/4) * 3^(5/4) * n^(3/4)). - Vaclav Kotesovec, Nov 16 2017

A263538 Expansion of 3 * a(q^2) * b(q^2) * c(q^2) / (b(q) * c(q)^2) in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 1, 6, 12, 5, 36, 60, 24, 150, 228, 86, 504, 732, 262, 1488, 2088, 725, 3996, 5460, 1852, 9972, 13344, 4436, 23472, 30876, 10103, 52644, 68268, 22040, 113364, 145224, 46336, 235734, 298800, 94378, 475488, 597108, 186926, 933672, 1162824, 361126, 1790028
Offset: 0

Views

Author

Michael Somos, Oct 20 2015

Keywords

Comments

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

Examples

			G.f. = 1 + x + 6*x^2 + 12*x^3 + 5*x^4 + 36*x^5 + 60*x^6 + 24*x^7 + 150*x^8 +  ...

Links

Crossrefs

Cf. A262930.

Programs

  • Mathematica
    a:= With[{nmax = 50}, CoefficientList[Series[(QPochhammer[x^2]^3 + 9*x^2*QPochhammer[x^18]^3)*QPochhammer[x^2]^2*QPochhammer[x^6]/ (QPochhammer[x]*QPochhammer[x^3]^5), {x, 0, nmax}], x]]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 31 2018 *)
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^3 + 9 * x^2 * eta(x^18 + A)^3) * eta(x^2 + A)^2 * eta(x^6 + A) / (eta(x + A) * eta(x^3 + A)^5), n))};

Formula

a(n) = A262930(2*n).
Showing 1-5 of 5 results.

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