A259065 E.g.f.: Series_Reversion( 5*x - 4*x*exp(x) ).

1, 8, 204, 8656, 514100, 39254904, 3663341724, 404021632928, 51413304278916, 7414832746025800, 1195170934203398636, 212923233266007511152, 41545466670049713766356, 8811212141812890158250776, 2018230889016461893216938300, 496523506149784085749952075584, 130578628540561635331879674437156

Offset

1,2

Links

G. C. Greubel, Table of n, a(n) for n = 1..305

Formula

O.g.f.: x * Sum_{n>=0} 4^n / (5 - n*x)^(n+1).

E.g.f.: x + Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^n / n!.

E.g.f.: x*exp( Sum_{n>=1} d^(n-1)/dx^(n-1) 4^n * (exp(x)-1)^n * x^(n-1) / n! ).

a(n) ~ (c/(5*exp(1)))^n * n^(n-1) / (sqrt(c+1) * (c-1)^(2*n-1)), where c = LambertW(5*exp(1)/4). - Vaclav Kotesovec, Jun 19 2015

Example

E.g.f.: A(x) = x + 8*x^2/2! + 204*x^3/3! + 8656*x^4/4! + 514100*x^5/5! +...
where A(5*x - 4*x*exp(x)) = x.
Also we have the related infinite series.
O.g.f.: F(x) = x + 8*x^2 + 204*x^3 + 8656*x^4 + 514100*x^5 + 39254904*x^6 +...
where F(x)/x = 1/5 + 4/(5-x)^2 + 4^2/(5-2*x)^3 + 4^3/(5-3*x)^4 + 4^4/(5-4*x)^5 +...

Mathematica

Rest[CoefficientList[InverseSeries[Series[5*x - 4*x*E^x, {x, 0, 20}], x], x] * Range[0, 20]!] (* Vaclav Kotesovec, Jun 19 2015 *)

Prog

(PARI) {a(n) = local(A=x); A = serreverse(5*x - 4*x*exp(x +x*O(x^n) )); n!*polcoeff(A, n)}
for(n=1, 20, 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, 4^m*(exp(x+x*O(x^n))-1)^m * x^m/m!)); n!*polcoeff(A, n)}
for(n=1, 25, print1(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, 4^m*(exp(x+x*O(x^n))-1)^m * x^(m-1)/m!)+x*O(x^n))); n!*polcoeff(A, n)}
for(n=1, 25, print1(a(n), ", "))

Crossrefs

Cf. A053492, A259063, A259064, A259066.

Sequence in context: A317631 A309904 A241224 * A204247 A063856 A367540

Adjacent sequences: A259062 A259063 A259064 * A259066 A259067 A259068

Keyword

nonn

Author

Paul D. Hanna, Jun 17 2015

Status

approved