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A218306
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E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(2*x)*cosh(2*x).
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10
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1, 2, 20, 392, 11648, 466112, 23517824, 1434077696, 102618951680, 8432793964544, 782753794531328, 81007725700038656, 9249066952457584640, 1154952975718091325440, 156588371428134115868672, 22908199202756436344963072, 3597006040171205977538822144
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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} 2*(3*n+2)^(n-1) * cosh((3*n+2)*x) * x^n/n!.
E.g.f.: A(x) = 1/2 + 1/2 * exp( 2*x - 2/3 * LambertW(-3*x * exp(3*x)) ).
a(n) = Sum_{k=0..n} (3*k+2)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + Sum_{k>=0} (3*k+2)^(k-1) * x^k/(1 - (3*k+2)*x)^(k+1). (End)
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EXAMPLE
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E.g.f.: A(x) = 1 + 2*x + 20*x^2/2! + 392*x^3/3! + 11648*x^4/4! + 466112*x^5/5! +...
where
A(x) = cosh(2*x) + 2*5^0*cosh(5*x)*x + 2*8^1*cosh(8*x)*x^2/2! + 2*11^2*cosh(11*x)*x^3/3! + 2*14^3*cosh(14*x)*x^4/4! + 2*17^4*cosh(17*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(3*X)*cosh(3*X)))); Egf=exp(2*R)*cosh(2*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*(3*k+2)^(k-1)*cosh((3*k+2)*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, A218303, A218304, A218305, A218307, A218308, A218309, A218310.
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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