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%I #38 Jul 10 2021 00:21:08
%S 1,0,1,2,6,24,135,930,7105,59192,549360,5746080,66713361,839528052,
%T 11308954657,163038260294,2520332282910,41640324943968,
%U 730119174449151,13507292654421390,263004450921933817,5385277610047242620,115775314245285797256,2606072891349667903152,61248210450060537498321
%N Expansion of the e.g.f. sqrt(-1 + 2 / (1 - x) / exp(x)).
%F E.g.f. y(x) satisfies y*y' = exp(-x)*x/(1-x)^2.
%F a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A014304(k-1)*A008306(n,k) for n > 0.
%F For all p prime, a(p) == -1 (mod p).
%F For n > 1, a(n) == 0 (mod (n-1)).
%F a(n) ~ 2 * n^n / exp(n + 1/2). - _Vaclav Kotesovec_, Jul 06 2021
%e sqrt(-1+2/(1-x)/exp(x)) = 1 + x^2/2! + 2*x^3/3! + 6*x^4/4! + 24*x^5/5! + 135*x^6/6! + 930*x^7/7! + 7105*x^8/8! + 59192*x^9/9! + ...
%e a(23) = Sum_{k=1..11} (-1)^(k-1)*A014304(k-1)*A008306(23,k) = 2606072891349667903152.
%e For k=1, (-1)^(1-1)*A014304(1-1)*A008306(23,1) == -1 (mod 23), because A014304(0) = 1 and A008306(23,1) = (23-1)!
%e For k>=2, (-1)^(k-1)*A014304(k-1)*A008306(23,k) == 0 (mod 23), because A008306(23,k) == 0 (mod 23), result a(23) == -1 (mod 23).
%e a(18) = Sum_{k=1..9} (-1)^(k-1)*A014304(k-1)*A008306(18,k) = 730119174449151.
%e a(18) == 0 (mod (18-1)), because for k >= 1, A008306(18,k) == 0 (mod 17).
%p A014304:= proc(n) option remember; `if`(n=0, 1, (-1)^n + add(binomial(n,k)*A014304(k)* A014304(n-k-1), k=0..n-1)) end:
%p 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 a:= n-> add(((-1)^(k-1)*A014304(k-1)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
%p # second program:
%p a := series(sqrt(-1+2/(1-x)/exp(x)), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
%t CoefficientList[Series[Sqrt[-1+2/(1-x)/E^x], {x, 0, 24}], x] * Range[0, 24]!
%o (PARI) my(x='x+O('x^30)); Vec(serlaplace(sqrt(-1 + 2 / (1 - x) / exp(x)))) \\ _Michel Marcus_, Jul 06 2021
%Y Cf. A000166, A008306, A014304, A327006, A345697, A345969, A346119.
%K nonn
%O 0,4
%A _Mélika Tebni_, Jul 06 2021
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