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A282046 From solution to a certain functional equation. 1
1, 6, 66, 852, 11874, 172860, 2586108, 39403704, 608191266, 9477587436, 148782847572, 2349371691528, 37276209494076, 593819849120712, 9492182832078888, 152184075600182352, 2446292788927524258, 39414680294672271756, 636376864032528135204, 10294159197891381551208 (list; graph; refs; listen; history; text; internal format)
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

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Last modified May 11 09:11 EDT 2021. Contains 343788 sequences. (Running on oeis4.)