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.

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A117410 Expansion of q^(-5/24) * eta(q^2)^3 / eta(q) in powers of q.

Original entry on oeis.org

1, 1, -1, 0, -1, -2, 1, -1, -1, 0, 1, 1, -1, 1, 0, 2, 1, 0, 0, -1, 2, 1, 0, -1, 0, -1, 0, -1, 1, 1, -3, 0, -1, -1, -1, 1, 0, 0, 0, -1, -2, 0, 1, 0, 1, 0, 1, 0, 0, -1, 2, -1, 0, 1, 1, 3, 0, -1, 0, 1, -1, 0, 1, 0, 0, 2, 0, 1, -1, 0, -2, -1, 1, 0, 0, -1, 0, 0, 1, -1, 0, -1, -1, -1, 0, -2, -1, 0, 2, 1, -2, 0, 1, -1, 0, -2, -1, 1, -1, 1, 0, 0, 0, 1, 0
Offset: 0

Views

Author

Michael Somos, Mar 13 2006

Keywords

Comments

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

Examples

			G.f. = 1 + x - x^2 - x^4 - 2*x^5 + x^6 - x^7 - x^8 + x^10 + x^11 - x^12 + x^13 + ...
G.f. = q^5 + q^29 - q^53 - q^101 - 2*q^125 + q^149 - q^173 - q^197 + q^245 + ...

Links

Crossrefs

Cf. A107034.

Programs

  • Maple
    # Uses EulerTransform from A358369.
a := EulerTransform(BinaryRecurrenceSequence(0, 1, -2)):
seq(a(n), n = 0..104); # Peter Luschny, Nov 17 2022
  • Mathematica
    a[ n_] := SeriesCoefficient[ QPochhammer[x^2]^3 / QPochhammer[ x], {x, 0, n}]; (* Michael Somos, Jan 31 2015 *)
  • PARI
    {a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^3 / eta(x + A), n))};
  • PARI
    q='q+O('q^99); Vec(eta(q^2)^3/eta(q)) \\ Altug Alkan, Apr 17 2018
  • Sage
    # uses[EulerTransform from A166861]
b = BinaryRecurrenceSequence(0, 1, -2)
a = EulerTransform(b)
print([a(n) for n in range(105)]) # Peter Luschny, Nov 17 2022

Formula

Expansion of psi(x)^2 * chi(-x) = f(-x)^2 / chi(-x)^3 = f(-x)^5 / phi(-x)^3 = f(-x^2)^2 / chi(-x) = f(-x^2)^3 / f(-x) = psi(x) * f(-x^2) = f(x) * f(-x^4) = phi(-x)^2 / chi(-x)^5 in powers of x where phi(), psi(), chi(), f() are Ramanujan theta functions. - Michael Somos, Jan 31 2015
Euler transform of period 2 sequence [ 1, -2, ...].
Given A = A0 + A1 + A2 + A3 + A4 is the 5-section, then 0 = A3 * A1^2 - A2 * A4^2.
Given A = A0 + A1 + A2 + A3 + A4 + A5 + A6 is the 7-section, then 0 = A0*A6 + A1*A5 + A2*A4 + 4*A3^2, A3 = x^10 * A(x^49).
G.f.: Product_{k>0} (1 + x^k) * (1 - x^(2*k))^2.
A107034(n) = (-1)^n * a(n).

A318027 Expansion of Product_{k>=1} 1/((1 - x^k)*(1 - x^(4*k))).

Original entry on oeis.org

1, 1, 2, 3, 6, 8, 13, 18, 29, 39, 57, 77, 112, 148, 205, 271, 372, 484, 647, 838, 1110, 1423, 1852, 2361, 3051, 3857, 4922, 6191, 7849, 9805, 12319, 15314, 19131, 23649, 29333, 36099, 44556, 54568, 66963, 81683, 99803, 121229, 147413, 178411, 216111, 260590, 314365, 377819, 454229
Offset: 0

Views

Author

Ilya Gutkovskiy, Aug 13 2018

Keywords

Comments

Convolution of A000041 and A035444.
Convolution of A000712 and A082303.
Convolution inverse of A107034.
Number of partitions of n if there are 2 kinds of parts that are multiples of 4.

Examples

			a(5) = 8 because we have [5], [4, 1], [4', 1], [3, 2], [3, 1, 1], [2, 2, 1], [2, 1, 1, 1] and [1, 1, 1, 1, 1].

Links

Crossrefs

Cf. A000041, A000712, A002512 (self-convolution), A002513, A035444, A082303, A100853, A107034, A318026, A318028.

Programs

  • Maple
    a:=series(mul(1/((1-x^k)*(1-x^(4*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^(4 k))), {k, 1, nmax}], {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[1/(QPochhammer[x] QPochhammer[x^4]), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Exp[Sum[x^k (1 + x^k + x^(2 k) + 2*x^(3 k))/(k (1 - x^(4 k))), {k, 1, nmax}]], {x, 0, nmax}], x]
Table[Sum[PartitionsP[k] PartitionsP[n - 4 k], {k, 0, n/4}], {n, 0, 48}]

Formula

G.f.: exp(Sum_{k>=1} x^k*(1 + x^k + x^(2*k) + 2*x^(3*k))/(k*(1 - x^(4*k)))).
a(n) ~ 5^(3/4) * exp(sqrt(5*n/6)*Pi) / (2^(13/4) * 3^(3/4) * n^(5/4)). - Vaclav Kotesovec, Aug 14 2018
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