A218309 E.g.f. A(x) satisfies A( x/(exp(3*x)*cosh(3*x)) ) = exp(4*x)*cosh(4*x).

1, 4, 56, 1264, 40640, 1711744, 89533184, 5607463936, 409621790720, 34218229227520, 3219000547131392, 336858779869020160, 38823224436435845120, 4886982191317154529280, 667188807538423365632000, 98200163047169655115350016, 15501781660715229538766815232

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 + 4*x + 56*x^2/2! + 1264*x^3/3! + 40640*x^4/4! + 1711744*x^5/5! +...
where
A(x) = cosh(2*x) + 4*5^0*cosh(5*x)*x + 4*8^1*cosh(8*x)*x^2/2! + 4*11^2*cosh(11*x)*x^3/3! + 4*14^3*cosh(14*x)*x^4/4! + 4*17^4*cosh(17*x)*x^5/5! +...
		

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218307, A218308, A218310.

Programs

PARI

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

From Seiichi Manyama, Apr 23 2024: (Start)

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

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

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