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 A136749 G.f.: Sum_{n>=0} arctanh(2^n*x)^n / n!, a power series in x with integer coefficients. 1
 1, 2, 8, 88, 2816, 285088, 96376832, 112173964160, 458290670993408, 6667221644498203136, 349410482551421802119168, 66605167708510907980664608768, 46557944823739673536754738305957888, 120169056821375322042225614651624227643392 (list; graph; refs; listen; history; text; internal format)
 OFFSET 0,2 COMMENTS 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). LINKS FORMULA a(n) = [y^n] sqrt((1+y)/(1-y))^(2^n) for n >= 0. a(n) = n!*[x^n] exp( 2^n*arctanh(x) ). G.f.: Sum_{n>=0} log( (1 + 2^n*x)/(1 - 2^n*x) )^n /(2^n*n!). EXAMPLE G.f.: A(x) = 1 + 2*x + 8*x^2 + 88*x^3 + 2816*x^4 + 285088*x^5 + 96376832*x^6 + ... where A(x) = 1 + arctanh(2*x) + arctanh(2^2*x)^2/2! + arctanh(2^3*x)^3/3! + arctanh(2^4*x)^4/4! + ... PROG (PARI) {a(n)=polcoeff(sqrt((1+x)/(1-x +x*O(x^n)))^(2^n), n)} (PARI) {a(n)=polcoeff(exp(2^n*atanh(x +x*O(x^n))), n)} (PARI) {a(n)=polcoeff(sum(k=0, n, atanh(2^k*x +x*O(x^n))^k/k!), n)} (PARI) {a(n)=polcoeff(sum(k=0, n, log((1+2^k*x)/(1-2^k*x +x*O(x^n)))^k/(2^k*k!)), n)} CROSSREFS Cf. A136559, A136647. Sequence in context: A141313 A009144 A132316 * A226321 A054955 A012299 Adjacent sequences:  A136746 A136747 A136748 * A136750 A136751 A136752 KEYWORD nonn AUTHOR Paul D. Hanna, Jan 21 2008 STATUS approved

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Last modified April 22 09:59 EDT 2021. Contains 343174 sequences. (Running on oeis4.)