A218303 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(x)*cosh(x).

1, 1, 6, 76, 1480, 39056, 1303904, 52716224, 2504292480, 136741146880, 8439125550592, 580959483530240, 44138582550333440, 3668643339883089920, 331143571990522060800, 32258185015683531587584, 3373221864252806213435392, 376881845889001869159759872

Offset: 0

Author

Paul D. Hanna, Oct 25 2012

Keywords

• nonn

Comments

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!.

Examples

			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! +...
		

Links

• Vaclav Kotesovec, Table of n, a(n) for n = 0..300

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218304, A218305, A218306, A218307, A218308, A218309, A218310.

Programs

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),", "))

Formula

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

From Seiichi Manyama, Apr 23 2024: (Start)

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)