OFFSET
0,3
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} (5*n+1)^(n-1) * cosh((5*n+1)*x) * x^n/n!.
From Seiichi Manyama, Apr 23 2024: (Start)
E.g.f.: A(x) = 1/2 + 1/2 * exp( x - 1/5 * LambertW(-5*x * exp(5*x)) ).
a(n) = 1/2 * Sum_{k=0..n} (5*k+1)^(n-1) * binomial(n,k) for n > 0.
G.f.: 1/2 + 1/2 * Sum_{k>=0} (5*k+1)^(k-1) * x^k/(1 - (5*k+1)*x)^(k+1). (End)
EXAMPLE
E.g.f.: A(x) = 1 + x + 12*x^2/2! + 364*x^3/3! + 17248*x^4/4! + 1118816*x^5/5! +...
where
A(x) = cosh(x) + 6^0*cosh(6*x)*x + 11^1*cosh(11*x)*x^2/2! + 16^2*cosh(16*x)*x^3/3! + 21^3*cosh(21*x)*x^4/4! + 26^4*cosh(26*x)*x^5/5! +...
PROG
(PARI) {a(n)=local(Egf=1, X=x+x*O(x^n), R=serreverse(x/(exp(5*X)*cosh(5*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, (5*k+1)^(k-1)*cosh((5*k+1)*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