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A168405 E.g.f.: Sum_{n>=0} arcsin(2^n*x)^n/n!. 2

%I

%S 1,2,16,520,66560,33882400,69055283200,564153087455360,

%T 18462510039810703360,2418626471936038215754240,

%U 1267795676362601991645220044800,2658560574070850656450883768752998400

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

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

%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 520*x^3/3! + 66560*x^4/4! + ...

%e A(x) = 1 + arcsin(2*x) + arcsin(4*x)^2/2! + arcsin(8*x)^3/3! + arcsin(16*x)^4/4! + ... + arcsin(2^n*x)^n/n! + ...

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

%e G(x) = 1 + x + x^2/2! + 2*x^3/3! + 5*x^4/4! + 20*x^5/5! + 85*x^6/6! + ... + A006228(n)*x^n/n! + ...

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

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

%Y Cf. A006228 (exp(arcsin x)), variants: A136632, A168402, A168403, A168404, A168406, A168407, A168408.

%K nonn

%O 0,2

%A _Paul D. Hanna_, Nov 25 2009

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Last modified January 29 01:41 EST 2020. Contains 331328 sequences. (Running on oeis4.)