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.
G.f.: A(x) = 1 - 2*x + 2*x^2 - 2*x^3 - 2*x^6 + 2*x^7 - 4*x^8 - 2*x^10 - 6*x^11 - ...
nmax = 48; CoefficientList[Series[(1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))/(1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])), {x, 0, nmax}], x]
nmax = 48; CoefficientList[Series[Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}] Sum[(-1)^k x^(k^2)/Product[(1 -
x^m), {m, 1, k}], {k, 0, nmax}] / Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}], {x, 0, nmax}], x]
G.f.: A(x) = 1 - x^2 - 2*x^3 - x^6 - 4*x^7 - 4*x^8 - 2*x^9 - 3*x^10 - 6*x^11 - 13*x^12 - ...
nmax = 45; CoefficientList[Series[1/((1/(1 + ContinuedFractionK[-x^k, 1, {k, 1, nmax}])) (1/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))), {x, 0, nmax}], x]
nmax = 45; CoefficientList[Series[Sum[(-1)^k x^(k^2)/Product[(1 - x^m), {m, 1, k}], {k, 0, nmax}] / (Sum[(-1)^k x^(k (k + 1))/Product[(1 - x^m), {m, 1, k}], {k, 0,
nmax}] Product[(1 - x^(5 k - 1)) (1 - x^(5 k - 4))/((1 - x^(5 k - 2)) (1 - x^(5 k - 3))), {k, 1, nmax}]), {x, 0, nmax}], x]
def s(k, m, n)
s = 0
(1..n).each{|i| s += i if n % i == 0 && i % k == m}
s
end
def A007325(n)
ary = [1]
a = [0] + (1..n).map{|i| s(5, 1, i) + s(5, 4, i) - s(5, 2, i) - s(5, 3, i)}
(1..n).each{|i| ary << (1..i).inject(0){|s, j| s - a[j] * ary[-j]} / i}
ary
end
def mul(f_ary, b_ary, m)
s1, s2 = f_ary.size, b_ary.size
ary = Array.new(s1 + s2 - 1, 0)
(0..s1 - 1).each{|i|
(0..s2 - 1).each{|j|
ary[i + j] += f_ary[i] * b_ary[j]
}
}
ary[0..m]
end
def A285441(n)
ary1 = A007325(n)
ary2 = Array.new(n + 1, 0)
(0..n / 2).each{|i| ary2[i * 2] = ary1[i]}
ary = [-1] + mul(ary1, mul(ary2, ary2, n), n)[0..-2]
mul(ary2, (0..n).map{|i| -ary[i]}, n)
end
p A285441(100)
nmax = 36; CoefficientList[Series[1/(1 + ContinuedFractionK[(x/(1 + ContinuedFractionK[x^k, 1, {k, 1, nmax}]))^k, 1, {k, 1, nmax}]), {x, 0, nmax}], x]
g[x_] := g[x] = QPochhammer[x, x^5] QPochhammer[x^4, x^5]/(QPochhammer[x^2, x^5] QPochhammer[x^3, x^5]); a[n_] := a[n] = SeriesCoefficient[g[x g[x]], {x, 0, n}]; Table[a[n], {n, 0, 36}]
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