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.

A112206 Coefficients of replicable function number "72b".

Original entry on oeis.org

1, 1, 0, 2, 2, 1, 2, 2, 3, 4, 4, 4, 7, 7, 6, 10, 11, 11, 14, 16, 17, 21, 22, 24, 32, 34, 34, 44, 49, 50, 60, 66, 72, 84, 90, 98, 117, 125, 132, 156, 171, 181, 206, 226, 245, 277, 298, 322, 369, 397, 422, 480, 522, 557, 620, 674, 728, 807, 868, 936, 1043, 1121, 1198
Offset: 0

Views

Author

Michael Somos, Aug 28 2005

Keywords

Comments

From Michael Somos, Oct 28 2019: (Start)
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution squared is A112173.
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = f(t) where q = exp(2 Pi i t).
Given G.f. A(x), then B(q) = q^(-1) * A(q^6) satisfies 0 = f(B(q), B(q^2), B(q^4)) where f(u, v, w) = 2 + (u^2 - v)*v*w^2 + (u^2 + v)*v^2.
(End)

Examples

			G.f. = 1 + x + 2*x^3 + 2*x^4 + x^5 + 2*x^6 + 2*x^7 + 3*x^8 + ...
G.f. = q^-1 + q^5 + 2*q^17 + 2*q^23 + q^29 + 2*q^35 + 2*q^41 + ...

Links

Crossrefs

Cf. A112173, A112175, A328789, A328790.

Programs

  • Mathematica
    nmax = 60; CoefficientList[Series[Product[(1 + x^k)*(1 + x^(3*k)) / ((1 + x^(2*k))*(1 + x^(6*k))), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Sep 08 2015 *)
eta[q_]:= q^(1/24)*QPochhammer[q]; h:= q^(1/6)*((eta[q^2]*eta[q^6])^2/(eta[q]*eta[q^3]*eta[q^4]*eta[q^12])); a:= CoefficientList[Series [h, {q,0,60}], q]; Table[a[[n]], {n,1,50}] (* G. C. Greubel, Jun 01 2018 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2] QPochhammer[ -x^3, x^6], {x, 0 ,n}]; (* Michael Somos, Oct 28 2019 *)
  • PARI
    q='q+O('q^50); h=((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))); Vec(h) \\ G. C. Greubel, Jun 01 2018
  • PARI
    {a(n) = my(A); if( n < 0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A) * eta(x^6 + A))^2 / (eta(x + A) * eta(x^3 + A) * eta(x^4 + A) * eta(x^12 + A)), n))}; /* Michael Somos, Oct 28 2019 */

Formula

a(n) ~ exp(sqrt(2*n)*Pi/3) / (2^(5/4) * sqrt(3) * n^(3/4)). - Vaclav Kotesovec, Sep 08 2015
Expansion of q^(1/6)*((eta(q^2)*eta(q^6))^2/(eta(q)*eta(q^3)*eta(q^4) *eta(q^12))) in powers of q. - G. C. Greubel, Jun 01 2018
From Michael Somos, Oct 28 2019: (Start)
Expansion of chi(x) * chi(x^3) in powers of x where chi() is a Ramanujan theta function.
Euler transform of period 12 sequence [1, -1, 2, 0, 1, -2, 1, 0, 2, -1, 1, 0, ...].
G.f.: Product_{k>=0} (1 + x^(2*k + 1)) * (1 + x^(6*k + 3)).
a(n) = (-1)^n * A112175(n). a(2*n) = A328789(n). a(2*n + 1) = A328790(n).
(End)

A328798 Expansion of 1 / (chi(-x) * chi(-x^3)) in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 1, 1, 3, 3, 4, 7, 8, 10, 16, 19, 23, 33, 39, 48, 65, 77, 93, 122, 144, 173, 220, 259, 309, 384, 451, 534, 653, 764, 899, 1085, 1264, 1479, 1765, 2048, 2385, 2820, 3260, 3778, 4432, 5105, 5891, 6864, 7879, 9056, 10491, 12002, 13744, 15839, 18064, 20616, 23648
Offset: 0

Views

Author

Michael Somos, Oct 28 2019

Keywords

Comments

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Convolution inverse is A112175, 2nd power is A102315, 3rd power is A229180, 6th power is A123653.
f(-1 / (216 t)) = 1/2 g(t) where q = exp(2 Pi i t) and g() is g.f. for A112175.

Examples

			G.f. = 1 + x + x^2 + 3*x^3 + 3*x^4 + 4*x^5 + 7*x^6 + 8*x^7 + ...
G.f. = q + q^7 + q^13 + 3*q^19 + 3*q^25 + 4*q^31 + 7*q^37 + ...

Links

Crossrefs

Cf. A102315, A112175, A123653, A229180.

Programs

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

Formula

Expansion of q^(-1/6) * eta(q^2) * eta(q^6) / (eta(q) * eta(q^3)) in powers of q.
Euler transform of period 6 sequence [1, 0, 2, 0, 1, 0, ...].
G.f.: Product_{k>=1} (1 + x^k)^(-1) * (1 + x^(3*k))^(-1).
a(n) ~ exp(2*Pi*sqrt(n)/3) / (4*sqrt(3)*n^(3/4)). - Vaclav Kotesovec, Oct 31 2019

A318026 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(3*k))).

Original entry on oeis.org

1, 1, 2, 4, 6, 9, 16, 22, 33, 50, 70, 98, 143, 193, 266, 368, 493, 659, 892, 1170, 1543, 2035, 2642, 3422, 4448, 5694, 7294, 9334, 11839, 14982, 18968, 23812, 29868, 37410, 46598, 57924, 71953, 88913, 109728, 135212, 165991, 203407, 248986, 303706, 369939, 449967, 545820, 661038, 799629
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

Comments

Convolution of A000041 and A035377.
Convolution of A000712 and A137569.
Convolution inverse of A030203.
Number of partitions of n if there are 2 kinds of parts that are multiples of 3.

Examples

			a(4) = 6 because we have [4], [3, 1], [3', 1], [2, 2], [2, 1, 1] and [1, 1, 1, 1].

Links

Crossrefs

Cf. A000041, A000712, A002513, A030203, A030206, A035377, A112175, A137569, A318027, A318028.

Programs

  • Maple
    a:=series(mul(1/((1-x^k)*(1-x^(3*k))),k=1..55),x=0,49): seq(coeff(a,x,n),n=0..48); # Paolo P. Lava, Apr 02 2019
  • Mathematica
    nmax = 48; CoefficientList[Series[Product[1/((1 - x^k) (1 - x^(3 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^3]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + 2 x^(2 k))/(k (1 - x^(3 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 3 k], {k, 0, n/3}], {n, 0, 48}]

Formula

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + 2*x^(2*k))/(k*(1 - x^(3*k)))).
a(n) ~ exp(2*sqrt(2*n)*Pi/3) / (3 * 2^(5/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018
Showing 1-3 of 3 results.

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