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A345697
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Expansion of the e.g.f. sqrt(1 / (2*exp(x) - 2*x*exp(x) - 1)).
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5
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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
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OFFSET
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0,4
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LINKS
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FORMULA
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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
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EXAMPLE
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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! + ...
For k>=2, A008306(13,k) == 0 (mod 13), result a(13) == -1 (mod 13).
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MAPLE
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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:
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);
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MATHEMATICA
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CoefficientList[Series[Sqrt[1/(2*E^x-2*x*E^x-1)], {x, 0, 24}], x] * Range[0, 24]!
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PROG
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(PARI) my(x='x+O('x^25)); Vec(serlaplace(sqrt(1 / (2*exp(x) - 2*x*exp(x) -1)))) \\ Michel Marcus, Jun 24 2021
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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