A295238 Expansion of e.g.f. 2/(1 + sqrt(1 - 4*x*exp(x))).

1, 1, 6, 57, 796, 14785, 344046, 9640225, 316255416, 11896233345, 504918768250, 23874754106401, 1244712973780068, 70940791877082049, 4388291507415513894, 292823509879910802465, 20966854494419642792176, 1603540841320336494905089, 130464295561360336835272050

Offset

0,3

Comments

Inverse binomial transform of A194471.

Links

Seiichi Manyama, Table of n, a(n) for n = 0..356

Formula

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

a(n) ~ sqrt(2*(1 + LambertW(1/4))) * n^(n-1) / ((LambertW(1/4))^n * exp(n)). - Vaclav Kotesovec, Nov 18 2017

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

Maple

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

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A032119 A294511 A242817 * A256016 A361291 A145170

Adjacent sequences: A295235 A295236 A295237 * A295239 A295240 A295241

Keyword

nonn

Author

Ilya Gutkovskiy, Nov 18 2017

Status

approved