A345697 Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).

1, 0, 1, 2, 12, 64, 485, 4038, 39991, 441992, 5492322, 75171700, 1127989577, 18381446004, 323527186957, 6114296752718, 123513004310640, 2655648779976640, 60554669008300565, 1459559515622280282, 37079264125376670955, 990226180225789628660, 27733277682719819190246, 812818183963966524137332, 24880254143735238825011057

Offset

0,4

Links

Table of n, a(n) for n=0..24.

Formula

E.g.f. y(x) satisfies y' = x*exp(x)*y^3.

a(0)=1, a(n) = Sum_{k=1..floor(n/2)} A014307(k)*A008306(n,k) for n >= 1.

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

a(n) ~ sqrt(2*c) * n^n / ((1-c)^(n+1) * exp(n)), where c = -LambertW(-exp(-1)/2). - Vaclav Kotesovec, Jun 25 2021

Example

sqrt(1/(2*exp(x)-2*x*exp(x)-1)) = 1 + x^2/2! + 2*x^3/3! + 12*x^4/4! + 64*x^5/5! + 485*x^6/6! + 4038*x^7/7! + 39991*x^8/8! + 441992*x^9/9! + ...
a(13) = Sum_{k=1..6} A014307(k)*A008306(13,k) = 18381446004.
A014307(1)*A008306(13,1) == -1 (mod 13), because A014307(1) = 1 and A008306(13,1) = (13-1)!
For k>=2, A008306(13,k) == 0 (mod 13), result a(13) == -1 (mod 13).

Maple

A014307 := proc(n) option remember; `if`(n=0, 1 , 1+add((-1+binomial(n, k))*A014307(k), k=1..n-1)) end:
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((A014307(k)*A008306(n, k)), k=1..floor(n/2)):a(0):=1 ;
seq(a(n), n=0..24);
# second program:
a := series(sqrt((1/(2*exp(x)-2*x*exp(x)-1))), x=0, 25):
seq(n!*coeff(a, x, n), n=0..24);

Mathematica

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

Prog

(PARI) my(x='x+O('x^25)); Vec(serlaplace(sqrt(1 / (2*exp(x) - 2*x*exp(x) -1)))) \\ Michel Marcus, Jun 24 2021

Crossrefs

Cf. A000166, A008306, A014307, A327006.

Sequence in context: A321989 A226226 A126737 * A097632 A076804 A190153

Adjacent sequences: A345694 A345695 A345696 * A345698 A345699 A345700

Keyword

nonn

Author

Mélika Tebni, Jun 24 2021

Status

approved