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 A168403 E.g.f.: Sum_{n>=0} sin(2^n*x)^n/n!. 3

%I #5 Oct 11 2020 04:05:50

%S 1,2,16,504,64512,33226784,68383997952,561747553419136,

%T 18430982918118572032,2417076909966155927519744,

%U 1267505531841541043488055885824,2658351411163282144153185664555284480

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

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

%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Oct 11 2020

%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 504*x^3/3! + 64512*x^4/4! +...

%e 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! +...

%e a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(sin(x)):

%e 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! +...

%t 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 *)

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

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

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

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 25 2009

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Last modified June 5 03:05 EDT 2023. Contains 363130 sequences. (Running on oeis4.)