A295238 - OEIS

%I #22 Oct 30 2024 10:27:33

%S 1,1,6,57,796,14785,344046,9640225,316255416,11896233345,504918768250,

%T 23874754106401,1244712973780068,70940791877082049,

%U 4388291507415513894,292823509879910802465,20966854494419642792176,1603540841320336494905089,130464295561360336835272050

%N Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

%C Inverse binomial transform of A194471.

%H Seiichi Manyama, <a href="/A295238/b295238.txt">Table of n, a(n) for n = 0..356</a>

%F E.g.f.: 1/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - x*exp(x)/(1 - ...))))), a continued fraction.

%F a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - _Vaclav Kotesovec_, Nov 18 2017

%F a(n) = n! * Sum_{k=0..n} k^(n-k) * binomial(2*k+1,k)/( (2*k+1)*(n-k)! ) = n! * Sum_{k=0..n} k^(n-k) * A000108(k)/(n-k)!. - _Seiichi Manyama_, Aug 15 2023

%p a:=series(2/(1+sqrt(1-4*x*exp(x))),x=0,19): seq(n!*coeff(a,x,n),n=0..18); # _Paolo P. Lava_, Mar 27 2019

%t nmax = 18; CoefficientList[Series[2/(1 + Sqrt[1 - 4 x Exp[x]]), {x, 0, nmax}], x] Range[0, nmax]!

%t nmax = 18; CoefficientList[Series[1/(1 + ContinuedFractionK[-x Exp[x], 1, {k, 1, nmax}]), {x, 0, nmax}], x] Range[0, nmax]!

%t Table[Sum[(-1)^(n - k) Binomial[n, k] k! Sum[(m + 1)^(k - m - 1) Binomial[2 m, m]/(k - m)!, {m, 0, k}], {k, 0, n}], {n, 0, 18}]

%o (PARI) a(n) = n!*sum(k=0, n, k^(n-k)*binomial(2*k, k)/((k+1)*(n-k)!)); \\ _Seiichi Manyama_, Aug 15 2023

%Y Cf. A000108, A006531, A052895, A194471, A213644, A295239.

%K nonn

%O 0,3

%A _Ilya Gutkovskiy_, Nov 18 2017