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.

A341243 Expansion of (-1 + Product_{k>=1} 1 / (1 + (-x)^k))^4.

Original entry on oeis.org

1, 0, 4, 4, 10, 16, 26, 44, 63, 100, 144, 212, 297, 420, 584, 796, 1081, 1452, 1940, 2556, 3355, 4372, 5668, 7288, 9327, 11892, 15076, 19012, 23884, 29904, 37276, 46284, 57276, 70680, 86918, 106528, 130220, 158784, 193054, 234076, 283178, 341824, 411616, 494512, 592933
Offset: 4

Views

Author

Ilya Gutkovskiy, Feb 07 2021

Keywords

Links

Crossrefs

Cf. A000700, A001482, A022599, A112160, A327382, A338463, A341222, A341241, A341244, A341245, A341246, A341247, A341251.
Column k=4 of A341279.

Programs

  • Maple
    g:= proc(n) option remember; `if`(n=0, 1, add(add([0, d, -d, d]
      [1+irem(d, 4)], d=numtheory[divisors](j))*g(n-j), j=1..n)/n)
    end:
b:= proc(n, k) option remember; `if`(k<2, `if`(n=0, 1-k, g(n)),
      (q-> add(b(j, q)*b(n-j, k-q), j=0..n))(iquo(k, 2)))
    end:
a:= n-> b(n, 4):
seq(a(n), n=4..48);  # Alois P. Heinz, Feb 07 2021
  • Mathematica
    nmax = 48; CoefficientList[Series[(-1 + Product[1/(1 + (-x)^k), {k, 1, nmax}])^4, {x, 0, nmax}], x] // Drop[#, 4] &

Formula

G.f.: (-1 + Product_{k>=1} (1 + x^(2*k - 1)))^4.
a(n) ~ A112160(n). - Vaclav Kotesovec, Feb 20 2021

A101127 McKay-Thompson series of class 12D for the Monster group.

Original entry on oeis.org

1, 8, 28, 64, 134, 288, 568, 1024, 1809, 3152, 5316, 8704, 13990, 22208, 34696, 53248, 80724, 121240, 180068, 264448, 384940, 556064, 796760, 1132544, 1598789, 2243056, 3127360, 4333568, 5971922, 8188096, 11170160, 15163392, 20491033
Offset: 0

Views

Author

Michael Somos, Dec 02 2004

Keywords

Comments

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

Examples

			T12D = 1/q + 8*q^2 + 28*q^5 + 64*q^8 + 134*q^11 + 288*q^14 + 568*q^17 + ...

References

  • D. Ford, J. McKay and S. P. Norton, More on replicable functions, Commun. Algebra 22, No. 13, 5175-5193 (1994).

Links

Crossrefs

cf. A007259, A112160.

Programs

  • Mathematica
    nmax = 50; CoefficientList[Series[Product[(1 + x^(2*k+1))^8, {k, 0, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 27 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ -x, x^2]^8, {x, 0, n}]; (* Michael Somos, Sep 12 2017 *)
  • PARI
    {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^2 / (eta(x + A) * eta(x^4 + A)))^8, n))};
  • PARI
    {a(n) = my(A); if(n<0, 0, A = x * O(x^n); polcoeff( prod(k=1, (n+1)\2, 1 + x^(2*k-1), 1 + A)^8, n))};

Formula

Expansion of q^(1/3) * (eta(q^2)^2 / (eta(q) * eta(q^4)))^8 in powers of q.
Euler transform of period 4 sequence [8, -8, 8, 0, ...].
Given g.f. A(x), B(q) = A(q^3) / q satisfies 0 = f(B(q), B(q^2)) where f(u, v) = u*v*(u^3+v^3) -(u*v)^3 + 15*(u*v)^2 - 32*u*v + 16.
G.f.: (Product_{k>0} (1 + x^(2*k-1)))^8.
A007259(n) = (-1)^n * a(n). Convolution square of A112160.
a(n) ~ exp(2*Pi*sqrt(n/3)) / (2 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Aug 27 2015
Expansion of chi(x)^8 in powers of x where chi() is a Ramanujan theta function. - Michael Somos, Sep 12 2017
G.f.: exp(8*Sum_{k>=1} x^k/(k*(1 - (-x)^k))). - Ilya Gutkovskiy, Jun 07 2018

A224916 Expansion of chi(x)^2 / chi(-x^2)^6 in powers of x where chi() is a Ramanujan theta function.

Original entry on oeis.org

1, 2, 7, 14, 31, 58, 112, 196, 347, 580, 966, 1554, 2485, 3872, 5993, 9102, 13719, 20384, 30068, 43836, 63481, 91048, 129763, 183448, 257839, 359862, 499583, 689312, 946416, 1292388, 1756838, 2376598, 3201557, 4293942, 5736736, 7633702, 10121408, 13370634
Offset: 0

Views

Author

Michael Somos, Apr 19 2013

Keywords

Comments

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

Examples

			1 + 2*x + 7*x^2 + 14*x^3 + 31*x^4 + 58*x^5 + 112*x^6 + 196*x^7 + 347*x^8 + ...
q^5 + 2*q^17 + 7*q^29 + 14*q^41 + 31*q^53 + 58*q^65 + 112*q^77 + 196*q^89 + ...

Links

Crossrefs

Cf. A098613, A112160.

Programs

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

Formula

Expansion of q^(-5/12) * (eta(q^4)^2 / (eta(q) * eta(q^2)))^2 in powers of q.
Expansion of psi(x^2)^2 / f(-x)^2 = 1 / (chi(-x)^2 * chi(-x^2)^4) = 1 / (chi(x)^4 * chi(-x)^6 ) in powers of x where psi(), chi(), f() are Ramanujan theta functions.
Expansion of (chi(x)^4 - chi(-x)^4) / (8*x) in powers of x^2 where chi() is a Ramanujan theta function.
Euler transform of period 4 sequence [ 2, 4, 2, 0, ...].
G.f.: Product_{k>0} (1 + x^k)^2 * (1 + x^(2*k))^4.
G.f.: (Sum_{k>0} x^(k^2 - k)) / (Product_{k>0} (1 - x^k))^2. - Michael Somos, Jul 04 2013
a(n) = A112160(2*n + 1) / 4.
Convolution square of A098613. - Michael Somos, Jul 04 2013
a(n) ~ exp(2*Pi*sqrt(n/3)) / (16 * 3^(1/4) * n^(3/4)). - Vaclav Kotesovec, Sep 07 2015

A227033 Expansion of (phi(x) / f(-x^4))^2 in powers of x where phi(), f() are Ramanujan theta functions.

Original entry on oeis.org

1, 4, 4, 0, 6, 16, 8, 0, 17, 40, 28, 0, 38, 96, 56, 0, 84, 204, 124, 0, 172, 400, 232, 0, 325, 760, 448, 0, 594, 1376, 784, 0, 1049, 2404, 1388, 0, 1796, 4096, 2320, 0, 3005, 6808, 3864, 0, 4912, 11072, 6216, 0, 7877, 17688, 9940, 0, 12430, 27792, 15488, 0
Offset: 0

Views

Author

Michael Somos, Jul 03 2013

Keywords

Comments

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

Examples

			G.f. = 1 + 4*x + 4*x^2 + 6*x^4 + 16*x^5 + 8*x^6 + 17*x^8 + 40*x^9 + 28*x^10 + ...
G.f. = 1/q + 4*q^2 + 4*q^5 + 6*q^11 + 16*q^14 + 8*q^17 + 17*q^23 + 40*q^26 + ...

Links

Crossrefs

Cf. A022569, A112160.

Programs

  • Mathematica
    a[ n_] := SeriesCoefficient[ (EllipticTheta[ 3, 0, x] / QPochhammer[ x^4])^2, {x, 0, n}];
  • PARI
    {a(n) = my(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))^2, n))};

Formula

Expansion of q^(1/3) * (eta(q^2)^5 / (eta(q)^2 * eta(q^4)^3))^2 in powers of q.
Euler transform of period 4 sequence [4, -6, 4, 0, ...].
a(4*n + 3) = 0. a(2*n) = A112160(n). a(4*n + 1) = 4 * A022569(n).
Showing 1-4 of 4 results.

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