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A218303
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E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(x)*cosh(x).
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11
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1, 1, 6, 76, 1480, 39056, 1303904, 52716224, 2504292480, 136741146880, 8439125550592, 580959483530240, 44138582550333440, 3668643339883089920, 331143571990522060800, 32258185015683531587584, 3373221864252806213435392, 376881845889001869159759872
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OFFSET
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0,3
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COMMENTS
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More generally, if A( x/(exp(t*x)*cosh(t*x)) ) = exp(m*x)*cosh(m*x),
then A(x) = Sum_{n>=0} m*(n*t+m)^(n-1) * cosh((n*t+m)*x) * x^n/n!.
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LINKS
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FORMULA
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E.g.f.: A(x) = Sum_{n>=0} (2*n+1)^(n-1) * cosh((2*n+1)*x) * x^n/n!.
a(n) ~ c * 2^n * n^(n-1) / (exp(n) * (LambertW(exp(-1)))^n), where c = sqrt(1 + LambertW(exp(-1)))/(4*sqrt(LambertW(exp(-1)))) = 0.535672560704567808218663129282561449... . - Vaclav Kotesovec, Jul 13 2014, updated Jun 10 2019
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/2 * LambertW(-2*x * exp(2*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (2*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (2*k+1)^(k-1) * x^k/(1 - (2*k+1)*x)^(k+1). (End)
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EXAMPLE
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E.g.f.: A(x) = 1 + x + 6*x^2/2! + 76*x^3/3! + 1480*x^4/4! + 39056*x^5/5! +...
where
A(x) = cosh(x) + 3^0*cosh(3*x)*x + 5^1*cosh(5*x)*x^2/2! + 7^2*cosh(7*x)*x^3/3! + 9^3*cosh(9*x)*x^4/4! + 11^4*cosh(11*x)*x^5/5! +...
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PROG
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(PARI) {a(n)=local(Egf=1, X=x+x*O(x^n), R=serreverse(x/(exp(2*X)*cosh(2*X)))); Egf=exp(R)*cosh(R); n!*polcoeff(Egf, n)}
for(n=0, 25, print1(a(n), ", "))
(PARI) /* Formula derived from a LambertW identity: */
{a(n)=local(Egf=1, X=x+x*O(x^n)); Egf=sum(k=0, n, (2*k+1)^(k-1)*cosh((2*k+1)*X)*x^k/k!); n!*polcoeff(Egf, n)}
for(n=0, 25, print1(a(n), ", "))
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CROSSREFS
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Cf. A201595, A218300, A218301, A218302, A218304, A218305, A218306, A218307, A218308, A218309, A218310.
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KEYWORD
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nonn,changed
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AUTHOR
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STATUS
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approved
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