A136647 G.f.: A(x) = Sum_{n>=0} arcsinh( 2^n*x )^n / n!; a power series in x with integer coefficients.

1, 2, 8, 84, 2688, 276892, 94978048, 111457917800, 457117679616000, 6660816097416169260, 349290546231751288553472, 66597307693046550483175282456, 46556113319179632622352835689840640

Offset

0,2

Links

G. C. Greubel, Table of n, a(n) for n = 0..50

Formula

a(n) = [y^n] ( sqrt(1+y^2) + y )^(2^n), since log(sqrt(1+y^2) + y) = arcsinh(y); [y^n] F(y) denotes the coefficient of y^n in F(y).

Example

G.f.: A(x) = 1 + 2*x + 8*x^2 + 84*x^3 + 2688*x^4 + 276892*x^5 + ...
This is a special application of the following identity.
Let F(x), G(x), be power series in x such that F(0)=1, G(0)=1, then
Sum_{n>=0} m^n * H(q^n*x) * log( F(q^n*x)*G(x) )^n / n! =
Sum_{n>=0} x^n * G(x)^(m*q^n) * [y^n] H(y)*F(y)^(m*q^n).

Maple

m:=30;
S:=series( add( arcsinh(2^j*x)^j/j! , j=0..m+2), x, m+1):
seq(coeff(S, x, j), j=0..m); # G. C. Greubel, Mar 15 2021

Mathematica

With[{m=30}, CoefficientList[Series[Sum[ArcSinh[2^j*x]^j/j!, {j, 0, m+2}], {x, 0, m}], x]] (* G. C. Greubel, Mar 15 2021 *)

Prog

(PARI) {a(n)=polcoeff(sum(k=0, n, asinh(2^k*x +x*O(x^n))^k/k!), n)}
(PARI) {a(n)=polcoeff((sqrt(1+x^2)+x+x*O(x^n))^(2^n), n)}
(Magma)
m:=30;
R<x>:=PowerSeriesRing(Rationals(), 30);
Coefficients(R!( (&+[Argsinh(2^j*x)^j/Factorial(j): j in [0..m+2]]) )); // G. C. Greubel, Mar 15 2021

Crossrefs

Cf. A136558.

Sequence in context: A295764 A261683 A134089 * A306001 A261730 A052456

Adjacent sequences: A136644 A136645 A136646 * A136648 A136649 A136650

Keyword

nonn

Author

Paul D. Hanna, Jan 20 2008

Status

approved