A218304 E.g.f. A(x) satisfies A( x/(exp(2*x)*cosh(2*x)) ) = exp(3*x)*cosh(3*x).

1, 3, 30, 468, 10248, 291888, 10282464, 432631104, 21195292800, 1186054914816, 74676568432128, 5226914768016384, 402722750814750720, 33876716756962652160, 3089713688099323502592, 303723970839738425622528, 32015024916407062538256384

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 + 3*x + 30*x^2/2! + 468*x^3/3! + 10248*x^4/4! + 291888*x^5/5! +...
where
A(x) = cosh(3*x) + 3*5^0*cosh(5*x)*x + 3*7^1*cosh(7*x)*x^2/2! + 3*9^2*cosh(9*x)*x^3/3! + 3*11^3*cosh(11*x)*x^4/4! + 3*13^4*cosh(13*x)*x^5/5! +...
		

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, 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(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*(2*k+3)^(k-1)*cosh((2*k+3)*X)*x^k/k!); n!*polcoeff(Egf,n)}
for(n=0,25,print1(a(n),", "))

Formula

E.g.f.: A(x) = Sum_{n>=0} 3*(2*n+3)^(n-1) * cosh((2*n+3)*x) * x^n/n!.

From Seiichi Manyama, Apr 23 2024: (Start)

E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/2 * LambertW(-2*x * exp(2*x)) ).

a(n) = 3/2 * Sum_{k=0..n} (2*k+3)^(n-1) * binomial(n,k) for n > 0.

G.f.: 1/2 + 3/2 * Sum_{k>=0} (2*k+3)^(k-1) * x^k/(1 - (2*k+3)*x)^(k+1). (End)