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Search: id:A136749
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| A136749 |
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G.f.: A(x) = Sum_{n>=0} atanh(2^n*x)^n / n!, a power series in x with integer coefficients. |
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2
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| 1, 2, 8, 88, 2816, 285088, 96376832, 112173964160, 458290670993408, 6667221644498203136, 349410482551421802119168, 66605167708510907980664608768, 46557944823739673536754738305957888
(list; graph; listen)
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OFFSET
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0,2
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FORMULA
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a(n) = [y^n] [(1+y)/(1-y)]^{2^(n-1)} for n>=0.
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EXAMPLE
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G.f.: A(x) = Sum_{n>=0} log( (1 + 2^n*x)/(1 - 2^n*x) )^n /(2^n*n!);
A(x) = 1 + 2*x + 8*x^2 + 88*x^3 + 2816*x^4 + 285088*x^5 + 96376832*x^6 +...
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).
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PROGRAM
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(PARI) {a(n)=polcoeff(((1+x)/(1-x +x*O(x^n)))^(2^(n-1)), 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)}
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CROSSREFS
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Cf. A136559; A136647.
Adjacent sequences: A136746 A136747 A136748 this_sequence A136750 A136751 A136752
Sequence in context: A141313 A009144 A132316 this_sequence A054955 A012299 A012295
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KEYWORD
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nonn
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AUTHOR
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Paul D. Hanna (pauldhanna(AT)juno.com), Jan 21 2008
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