A141368 G.f.: Sum_{n>=0} arctanh(4^n*x)^n/n!, a power series in x having only integer coefficients.

1, 4, 128, 43712, 178978816, 9382678180864, 6558857974821945344, 62879510456046477909016576, 8439543050458648574249946721550336, 16110026905906831711301708576024644666261504

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..40

Formula

a(n) = [x^n] [ sqrt((1+x)/(1-x)) ]^(4^n).

More generally, the following coefficient of x^n in the series:

[x^n] Sum_{n>=0} arctanh(q^n*x)^n/n! = [x^n] [ sqrt((1+x)/(1-x)) ]^(q^n) is an integer for any even integer q.

Example

G.f. A(x) = 1 + 4*x + 128*x^2 + 43712*x^3 + 178978816*x^4 + ...

Mathematica

Table[SeriesCoefficient[(Sqrt[(1 + x)/(1 - x)])^(4^n), {x, 0, n}], {n, 0, 25}] (* G. C. Greubel, Apr 15 2017 *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, n, atanh(4^k*x +x*O(x^n))^k/k!), n)}
(PARI) {a(n)=polcoeff(((1+x)/(1-x +x*O(x^n)))^(4^n/2), n)}

Crossrefs

Cf. A136749, A141367.

Sequence in context: A000318 A229385 A141367 * A146555 A305568 A057134

Adjacent sequences: A141365 A141366 A141367 * A141369 A141370 A141371

Keyword

nonn

Author

Paul D. Hanna, Jul 02 2008

Status

approved