OFFSET
0,2
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!.
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)
EXAMPLE
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! +...
PROG
(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), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Oct 25 2012
STATUS
approved