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%I #5 Oct 11 2020 04:08:56
%S 1,2,16,496,63488,32899584,68048945152,560546063546368,
%T 18415227534711980032,2416302301874471529480192,
%U 1267360472364492854214927581184,2658246833130441862918267501949222912
%N E.g.f.: Sum_{n>=0} tanh(2^n*x)^n/n!.
%F a(n) = [x^n/n! ] exp(2^n*tanh(x)).
%F a(n) ~ 2^(n^2). - _Vaclav Kotesovec_, Oct 11 2020
%e E.g.f.: A(x) = 1 + 2*x + 16*x^2/2! + 496*x^3/3! + 63488*x^4/4! +...
%e A(x) = 1 + tanh(2*x) + tanh(4*x)^2/2! + tanh(8*x)^3/3! + tanh(16*x)^4/4! +...+ tanh(2^n*x)^n/n! +...
%e a(n) = coefficient of x^n/n! in G(x)^(2^n) where G(x) = exp(tanh(x)):
%e G(x) = 1 + x + x^2/2! - x^3/3! - 7*x^4/4! - 3*x^5/5! + 97*x^6/6! + 275*x^7/7! +...+ A003723(n)*x^n/n! +...
%t nmax = 15; CoefficientList[Series[Sum[Tanh[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,tanh(2^k*x +x*O(x^n))^k/k!),n)}
%o (PARI) {a(n)=n!*polcoeff(exp(2^n*tanh(x +x*O(x^n))),n)}
%Y Cf. A003723 (exp(tanh x)), variant: A136632.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Nov 25 2009
%E Example corrected by _Paul D. Hanna_, Nov 25 2009
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