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.

A098613 Expansion of psi(x^2) / f(-x) in powers of x where psi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 1, 3, 4, 7, 10, 17, 23, 35, 48, 69, 93, 131, 173, 236, 310, 413, 536, 704, 903, 1170, 1489, 1904, 2403, 3044, 3811, 4784, 5951, 7409, 9157, 11325, 13912, 17095, 20891, 25519, 31029, 37708, 45632, 55184, 66495, 80050, 96064, 115173, 137680, 164425, 195860
Offset: 0

Views

Author

Michael Somos, Sep 17 2004

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
This sequence convolved with A000009 gives A001936. - Gary W. Adamson, Mar 24 2011
a(n) is the number of partitions of n in which each odd part can occur any number of times but each even part is of two kinds and each kind can occur at most once. - Michael Somos, Dec 01 2019

Examples

			G.f. = 1 + x + 3*x^2 + 4*x^3 + 7*x^4 + 10*x^5 + 17*x^6 + 23*x^7 + ...
G.f. = q^5 + q^29 + 3*q^53 + 4*q^77 + 7*q^101 + 10*q^125 + 17*q^149 + ...

Links

Crossrefs

Cf. A029552, A083906, A143161.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ EllipticTheta[ 2, 0, x] / (2 x^(1/4) QPochhammer[ x]), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
a[ n_] := SeriesCoefficient[ 1 / (QPochhammer[ x, x^2] QPochhammer[ x^2, x^4]^2), {x, 0, n}]; (* Michael Somos, Oct 29 2013 *)
nmax = 40; CoefficientList[ Series[Product[(1 + x^k) * (1 + x^(2*k))^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^2, x^2]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x, -x]^2 QPochhammer[ -x, x]^3, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x] QPochhammer[ -x^2, x^2]^2, {x, 0, n}]; (* Michael Somos, Sep 07 2015 *)
  • PARI
    {a(n) = if( n<0, 0, polcoeff( sum(k=0, (sqrtint(4*n+1)-1)\2, x^(k^2+k)) / eta(x + x * O(x^n)), n))};
  • PARI
    {a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^4 + A)^2 / (eta(x + A) * eta(x^2 + A)), n))};
  • PARI
    {a(n) = if( n<0,0, polcoeff( sum(k=0, 2*n+1, prod(i=1, k, (1 - x^(2*n+2 - i)) / (1 - x^i))) / 2, n^2))};

Formula

Expansion of chi(x) / chi(-x^2)^3 = 1 / (chi(-x)* chi(-x^2)^2) = 1 / (chi(x)^2 * chi(-x)^3) in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 07 2015
Expansion of q^(-5/24) * eta(q^4)^2 / (eta(q) * eta(q^2)) in powers of q.
Euler transform of period 4 sequence [1, 2, 1, 0, ...].
G.f. A(x) is the limit of x^(n^2+n) * P_{2*n+1}(1/x)/2 where P_n(q) = Sum_{k=0..n} C(n, k; q) and C(n, k; q) is the q-binomial coefficients. See A083906 for P_n(q).
G.f.: (Sum_{k>0} x^(k^2-k)) / (Product_{k>0} (1 - x^k)).
a(n) ~ exp(Pi*sqrt(2*n/3)) / (2^(11/4) * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 8^(-1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. for A143161. - Michael Somos, Sep 07 2015
G.f.: Product_{k>=1} (1 + x^(2*k))^2 / (1 - x^(2*k-1)). - Michael Somos, Dec 01 2019

A225853 Expansion of phi(x) / f(-x^4) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 2, 0, 0, 3, 2, 0, 0, 4, 6, 0, 0, 7, 8, 0, 0, 13, 14, 0, 0, 19, 20, 0, 0, 29, 34, 0, 0, 43, 46, 0, 0, 62, 70, 0, 0, 90, 96, 0, 0, 126, 138, 0, 0, 174, 186, 0, 0, 239, 262, 0, 0, 325, 346, 0, 0, 435, 472, 0, 0, 580, 620, 0, 0, 769, 826, 0, 0, 1007, 1072, 0
Offset: 0

Views

Author

Michael Somos, May 17 2013

Keywords

Comments

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

Examples

			1 + 2*x + 3*x^4 + 2*x^5 + 4*x^8 + 6*x^9 + 7*x^12 + 8*x^13 + 13*x^16 + ...
1/q + 2*q^5 + 3*q^23 + 2*q^29 + 4*q^47 + 6*q^53 + 7*q^71 + 8*q^77 + 13*q^95 + ...

Links

Crossrefs

Cf. A029552, A143161.

Programs

  • Mathematica
    a[n_]:= SeriesCoefficient[EllipticTheta[3,0,q]/QPochhammer[q^4],{q,0,n}];
a[n_]:= SeriesCoefficient[QPochhammer[q^2,q^4]^3/QPochhammer[q,q^2]^2, {q,0,n}];
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 / (eta(x + A)^2 * eta(x^4 + A)^3), n))}

Formula

Expansion of chi(x)^2 * chi(-x^2) = chi(x)^3 * chi(-x) = chi(-x^2)^3 / chi(-x)^2 in powers of x where chi() is a Ramanujan theta function.
Expansion of q^(1/4) * eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3) in powers of q.
Euler transform of period 4 sequence [ 2, -3, 2, 0, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (576 t)) = 2^(1/2) g(t) where q = exp(2 Pi i t) and g() is the g.f. of A029552.
G.f.: Product_{k>0} (1 - x^(4*k-2))^3 / (1 - x^(2*k-1))^2 = (Sum_{k in Z} x^k^2) / (Product_{k>0} (1 - x^(4*k))).
a(n) = (-1)^n * A143161(n). a(4*n + 2) = a(4*n + 3) = 0.

A259657 Expansion of phi(-x^3) * f(-x^4)^3 / f(-x^12) in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 0, 0, -2, -3, 0, 0, 6, 0, 0, 0, 0, 8, 0, 0, -12, -9, 0, 0, 6, 0, 0, 0, 0, 12, 0, 0, -2, -12, 0, 0, 18, 0, 0, 0, 0, 6, 0, 0, -24, -12, 0, 0, 6, 0, 0, 0, 0, 20, 0, 0, -12, -12, 0, 0, 24, 0, 0, 0, 0, 24, 0, 0, -12, -21, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, -14, -24
Offset: 0

Views

Author

Michael Somos, Jul 02 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 - 2*x^3 - 3*x^4 + 6*x^7 + 8*x^12 - 12*x^15 - 9*x^16 + 6*x^19 + ...

Links

Crossrefs

Cf. A014453, A143161, A259655.

Programs

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

Formula

Expansion of phi(-x^3) * b(x^4) in powers of x where phi() is a Ramanujan theta function and b() is a cubic AGM theta function.
Expansion of eta(q^3)^2 * eta(q^4)^3 / (eta(q^6) * eta(q^12)) in powers of q.
Euler transform of period 12 sequence [ 0, 0, -2, -3, 0, -1, 0, -3, -2, 0, 0, -3, ...].
G.f. is a period 1 Fourier series which satisfies f(-1 / (72 t)) = 72^(3/2) (t/i)^(3/2) g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A259655.
G.f.: Product_{k>0} (1 - x^(4*k))^3 / ((1 + x^(3*k))^2 * (1 + x^(6*k))).
a(3*n + 1) = -3 * A143161(n-1). a(3*n + 2) = a(4*n + 1) = a(4*n + 2) = 0. a(12*n) = A014453(n).
Showing 1-3 of 3 results.

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