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A218301
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E.g.f. A(x) satisfies A( x/(exp(x)*cosh(x)) ) = exp(3*x)*cosh(3*x).
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10
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1, 3, 24, 252, 3360, 55008, 1074816, 24499968, 639744000, 18856765440, 619897847808, 22502300590080, 894419152404480, 38651030120693760, 1804765006764441600, 90574514900736933888, 4862862027933962207232, 278158492957848901779456, 16889663645642083220324352
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OFFSET
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0,2
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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} 3*(n+3)^(n-1) * cosh((n+3)*x) * x^n/n!.
E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3*LambertW(-x * exp(x)) ).
a(n) = 3/2 * Sum_{k=0..n} (k+3)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 3/2 * Sum_{k>=0} (k+3)^(k-1) * x^k/(1 - (k+3)*x)^(k+1). (End)
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EXAMPLE
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E.g.f.: A(x) = 1 + 3*x + 24*x^2/2! + 252*x^3/3! + 3360*x^4/4! + 55008*x^5/5! +...
where
A(x) = cosh(3*x) + 3*4^0*cosh(4*x)*x + 3*5^1*cosh(5*x)*x^2/2! + 3*6^2*cosh(6*x)*x^3/3! + 3*7^3*cosh(7*x)*x^4/4! + 3*8^4*cosh(8*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(X)*cosh(X)))); Egf=exp(3*R)*cosh(3*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, 3*(k+3)^(k-1)*cosh((k+3)*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, A218302, A218303, 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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