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

%I A345969 #15 Jul 03 2021 02:08:56
%S A345969 1,0,1,2,18,104,1015,9666,116557,1504856,22300704,358916480,
%T A345969 6373675825,122332173300,2540560235161,56558354414870,
%U A345969 1346402030278050,34093192112537888,915570658175517151,25983157665663651150,777141557158947654637,24430880483991543481580
%N A345969 Expansion of the e.g.f. 1 / sqrt(3 - 2 / ((1 - x)*exp(x))).
%F A345969 E.g.f. y(x) satisfies y' = exp(-x)*y^3*x/(1-x)^2.
%F A345969 a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A305404(k)*A008306(n,k) for n > 0.
%F A345969 For all p prime, a(p) == -1 (mod p).
%F A345969 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
%e A345969 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! + ...
%e A345969 a(17) = Sum_{k=1..8} A305404(k)*A008306(17,k) = 34093192112537888.
%e A345969 For k=1, A305404(1)*A008306(17,1) == -1 (mod 17), because A305404(1) = 1 and A008306(17,1) = (17-1)!
%e A345969 For k>=2, A305404(k)*A008306(17,k) == 0 (mod 17), because A008306(17,k) == 0 (mod 17), result a(17) == -1 (mod 17).
%p A345969 A305404:= n-> add(Stirling2(n,k)*doublefactorial(2*k-1), k=0..n):
%p A345969 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:
%p A345969 a := n-> add((A305404(k)*A008306(n, k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
%p A345969 # second program:
%p A345969 a := series(1/sqrt(3-2/((1-x)*exp(x))), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
%t A345969 CoefficientList[Series[1/Sqrt[3-2/((1-x)*E^x)], {x, 0, 24}], x] * Range[0, 24]!
%o A345969 (PARI) my(x='x+O('x^30)); Vec(serlaplace(1/sqrt(3 - 2 / ((1 - x)*exp(x))))) \\ _Michel Marcus_, Jul 01 2021
%Y A345969 Cf. A000166, A008306, A305404, A327006, A345697.
%K A345969 nonn
%O A345969 0,4
%A A345969 _Mélika Tebni_, Jul 01 2021