A214363 E.g.f. satisfies: A(x) = x + A(x)^2 * cosh(A(x))^2 / 2.

1, 1, 3, 27, 285, 3585, 56595, 1062131, 22868685, 557624745, 15204727395, 458112683787, 15113457195837, 541914801559313, 20984168325697395, 872681528769576675, 38793582477781496685, 1835683831177469267769, 92124361183712633639235, 4887330703061330205124475

Offset

1,3

Comments

a(n) (mod 3) yields period 6 sequence: [0,0,0,0,0,2] starting at n=3.

a(6*n+2) == 2 (mod 3) for n>=1.

Links

Table of n, a(n) for n=1..20.

Formula

E.g.f. satisfies:

(1) A(x - x^2*cosh(x)^2/2) = x.

(2) A(x) = x + Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^(2*n)*x^(2*n) / (2^n*n!).

(2) A(x) = x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) cosh(x)^(2*n)*x^(2*n-1) / (2^n*n!) ).

a(n) ~ n^(n-1) * sqrt(2/(1 + (1+2*s^2)*cosh(2*s) + 4*s*sinh(2*s))) / (exp(n) * r^(n-1/2)), where s = 0.568824148293580379787367453... is the root of the equation s*cosh(s)*(cosh(s) + s*sinh(s)) = 1, and r = s - s^2*(cosh(s))^2/2 = 0.3488028982491643456675... - Vaclav Kotesovec, Jan 12 2014

Example

E.g.f: A(x) = x + x^2/2! + 3*x^3/3! + 27*x^4/4! + 285*x^5/5! + 3585*x^6/6! +...
Related expansions:
A(x)^2 = 2*x^2/2! + 6*x^3/3! + 30*x^4/4! + 330*x^5/5! + 4410*x^6/6! + 67830*x^7/7! +...
A(x) = x + cosh(x)^2*x^2/2 + d/dx cosh(x)^4*x^4/(4*2!) + d^2/dx^2 cosh(x)^6*x^6/(8*3!) + d^3/dx^3 cosh(x)^8*x^8/(16*4!) +...
log(A(x)/x) = 1 + cosh(x)^2*x/2 + d/dx cosh(x)^4*x^3/(4*2!) + d^2/dx^2 cosh(x)^6*x^5/(8*3!) + d^3/dx^3 cosh(x)^8*x^7/(16*4!) +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[x - (x^2*Cosh[x]^2)/2, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jan 12 2014 *)

Prog

(PARI) {a(n)=n!*polcoeff(serreverse(x-x^2/2*cosh(x+x*O(x^n))^2), n)}
for(n=1, 30, print1(a(n), ", "))
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x); A=x+sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^(2*m)*x^(2*m)/2^m/m!)); n!*polcoeff(A, n)}
(PARI) {Dx(n, F)=local(D=F); for(i=1, n, D=deriv(D)); D}
{a(n)=local(A=x+x^2+x*O(x^n)); A=x*exp(sum(m=1, n, Dx(m-1, cosh(x+x*O(x^n))^(2*m)*x^(2*m-1)/2^m/m!)+x*O(x^n))); n!*polcoeff(A, n)}

Crossrefs

Cf. A213643.

Sequence in context: A355363 A011544 A127503 * A204821 A365156 A200903

Adjacent sequences: A214360 A214361 A214362 * A214364 A214365 A214366

Keyword

nonn

Author

Paul D. Hanna, Jul 13 2012

Status

approved