A218308 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(3*x)*cosh(3*x).

1, 3, 42, 1116, 44616, 2394288, 161719200, 13187258304, 1261037553792, 138415816348416, 17155627044653568, 2370099000682257408, 361171910376568571904, 60185513513709805350912, 10887989148395358662270976, 2125192867898778619536457728

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 + 42*x^2/2! + 1116*x^3/3! + 44616*x^4/4! + 2394288*x^5/5! +...
where
A(x) = cosh(3*x) + 3*7^0*cosh(7*x)*x + 3*11^1*cosh(11*x)*x^2/2! + 3*15^2*cosh(15*x)*x^3/3! + 3*19^3*cosh(19*x)*x^4/4! + 3*23^4*cosh(23*x)*x^5/5! +...
		

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218309, A218310.

Programs

PARI

{a(n)=local(Egf=1,X=x+x*O(x^n),R=serreverse(x/(exp(4*X)*cosh(4*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*(4*k+3)^(k-1)*cosh((4*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} (4*n+1)^(n-1) * cosh((4*n+1)*x) * x^n/n!.

From Seiichi Manyama, Apr 23 2024: (Start)

E.g.f.: A(x) = 1/2 + 1/2 * exp( 3*x - 3/4 * LambertW(-4*x * exp(4*x)) ).

a(n) = 3/2 * Sum_{k=0..n} (4*k+3)^(n-1) * binomial(n,k) for n > 0.

G.f.: 1/2 + 3/2 * Sum_{k>=0} (4*k+3)^(k-1) * x^k/(1 - (4*k+3)*x)^(k+1). (End)