A282046 From solution to a certain functional equation.

1, 6, 66, 852, 11874, 172860, 2586108, 39403704, 608191266, 9477587436, 148782847572, 2349371691528, 37276209494076, 593819849120712, 9492182832078888, 152184075600182352, 2446292788927524258, 39414680294672271756, 636376864032528135204, 10294159197891381551208

Offset

0,2

Links

Table of n, a(n) for n=0..19.

Shaun Cooper, Jesús Guillera, Armin Straub, Wadim Zudilin, Crouching AGM, Hidden Modularity, arXiv:1604.01106 [math.NT], 5-April-2016. See Section 2.

Mathematica

terms = 20;
f[_] = 1;
Do[f[z_] = f[z] - f[z/(1+2z)^3]/(1+2z)^2 + f[z^2/(1-4z)^3]/(1-4z)^2 + O[z]^terms // Normal, {terms}];
g[z_] = Sqrt[f[z]] + O[z]^terms;
CoefficientList[g[z], z] (* Jean-François Alcover, Oct 10 2018, after Andrey Zabolotskiy *)

Prog

(Python)
fact = [1]
for i in range(1, 80):
    fact.append(fact[-1]*i)
def cnk(n, k):
    return fact[n]//fact[k]//fact[n-k]
def cc(N): # returns first N terms of A282045
    c = [1]
    for n in range(1, N):
        c.append(sum(cnk(n+k+1, 3*k+1)*4**(n-2*k)*c[k] for k in range(n//2+1)) - sum(cnk(n+2*k+1, 3*k+1)*(-2)**(n-k)*c[k] for k in range(n)))
    return c
def dd(ccc):
    d = [1]
    for n in range(1, len(ccc)):
        d.append((ccc[n]-sum(d[k]*d[n-k] for k in range(1, n)))//2)
    return d
print(dd(cc(25)))
# Andrey Zabolotskiy, Feb 22 2017

Crossrefs

Cf. A282045.

Sequence in context: A173535 A267141 A004355 * A124862 A130977 A191096

Adjacent sequences: A282043 A282044 A282045 * A282047 A282048 A282049

Keyword

nonn

Author

N. J. A. Sloane, Feb 21 2017

Extensions

More terms from Andrey Zabolotskiy, Feb 22 2017

Status

approved