A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).

1, 0, 1, 2, 18, 104, 1015, 9666, 116557, 1504856, 22300704, 358916480, 6373675825, 122332173300, 2540560235161, 56558354414870, 1346402030278050, 34093192112537888, 915570658175517151, 25983157665663651150, 777141557158947654637, 24430880483991543481580

Offset: 0

Author

Mélika Tebni, Jul 01 2021

Keywords

• nonn

Examples

			1/sqrt(3-2/((1-x)*exp(x))) =  1 + x^2/2! + 2*x^3/3! + 18*x^4/4! + 104*x^5/5! + 1015*x^6/6! + 9666*x^7/7! + 116557*x^8/8! + 1504856*x^9/9! + ...
a(17) = Sum_{k=1..8} A305404(k)*A008306(17,k) = 34093192112537888.
For k=1, A305404(1)*A008306(17,1) == -1 (mod 17), because A305404(1) = 1 and A008306(17,1) = (17-1)!
For k>=2, A305404(k)*A008306(17,k) == 0 (mod 17), because A008306(17,k) == 0 (mod 17), result a(17) == -1 (mod 17).
		

Crossrefs

Cf. A000166, A008306, A305404, A327006, A345697.

Programs

Maple

A305404:= n-> add(Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
A008306 := proc(n, k): if k=1 then (n-1)! ; elif n<=2*k-1 then 0; else (n-1)*procname(n-1, k)+(n-1)*procname(n-2, k-1) ; end if; end proc:
a := n-> add((A305404(k)*A008306(n, k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
# second program:
a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);

Mathematica

CoefficientList[Series[1/Sqrt[3-2/((1-x)*E^x)], {x, 0, 24}], x] * Range[0, 24]!

PARI

my(x='x+O('x^30)); Vec(serlaplace(1/sqrt(3 - 2 / ((1 - x)*exp(x))))) \\ Michel Marcus, Jul 01 2021

Formula

E.g.f. y(x) satisfies y' = exp(-x)*y^3*x/(1-x)^2.

a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A305404(k)*A008306(n,k) for n > 0.

For all p prime, a(p) == -1 (mod p).

a(n) ~ sqrt(-2*LambertW(-2*exp(-1)/3)/3) * n^n / (exp(n) * (1 + LambertW(-2*exp(-1)/3))^(n+1)). - Vaclav Kotesovec, Jul 01 2021