A218307 E.g.f. A(x) satisfies A( x/(exp(4*x)*cosh(4*x)) ) = exp(x)*cosh(x).

1, 1, 10, 244, 9288, 483216, 31949216, 2564959552, 242374510720, 26355555496192, 3241906046249472, 445085008158569472, 67469834196870809600, 11192986206960277688320, 2017105871358529382883328, 392394481517424330142203904, 81955683182673295403291541504

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!.

Links

Table of n, a(n) for n=0..16.

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( x - 1/4 * LambertW(-4*x * exp(4*x)) ).

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

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

Example

E.g.f.: A(x) = 1 + x + 10*x^2/2! + 244*x^3/3! + 9288*x^4/4! + 483216*x^5/5! +...
where
A(x) = cosh(x) + 5^0*cosh(5*x)*x + 9^1*cosh(9*x)*x^2/2! + 13^2*cosh(13*x)*x^3/3! + 17^3*cosh(17*x)*x^4/4! + 21^4*cosh(21*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(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, (4*k+1)^(k-1)*cosh((4*k+1)*X)*x^k/k!); n!*polcoeff(Egf, n)}
for(n=0, 25, print1(a(n), ", "))

Crossrefs

Cf. A201595, A218300, A218301, A218302, A218303, A218304, A218305, A218306, A218308, A218309, A218310.

Sequence in context: A331611 A122836 A116878 * A249564 A034222 A197437

Adjacent sequences: A218304 A218305 A218306 * A218308 A218309 A218310

Keyword

nonn

Author

Paul D. Hanna, Oct 25 2012

Status

approved