A168403 E.g.f.: Sum_{n>=0} sin(2^n*x)^n/n!.

1, 2, 16, 504, 64512, 33226784, 68383997952, 561747553419136, 18430982918118572032, 2417076909966155927519744, 1267505531841541043488055885824, 2658351411163282144153185664555284480

Offset: 0

Author

Paul D. Hanna, Nov 25 2009

Keywords

• nonn

Examples

			E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 504*x^3/3! + 64512*x^4/4! +...
A(x) = 1 + sin(2*x) + sin(4*x)^2/2! + sin(8*x)^3/3! + sin(16*x)^4/4! +...+ sin(2^n*x)^n/n! +...
a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(sin(x)):
G(x) = 1 + x + x^2/2! - 3*x^4/4! - 8*x^5/5! - 3*x^6/6! + 56*x^7/7! +...+ A002017(n)*x^n/n! +...
		

Crossrefs

Cf. A002017 (exp(sin x)), variants: A168402, A136632.

Programs

Mathematica

nmax = 12; CoefficientList[Series[Sum[Sin[2^k*x]^k/k!, {k, 0, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Oct 11 2020 *)

PARI

{a(n)=n!*polcoeff(sum(k=0,n,sin(2^k*x +x*O(x^n))^k/k!),n)}

PARI

{a(n)=n!*polcoeff(exp(2^n*sin(x +x*O(x^n))),n)}

Formula

a(n) = [x^n/n! ] exp(2^n*sin(x)) for n>=0.

a(n) ~ 2^(n^2). - Vaclav Kotesovec, Oct 11 2020