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A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).

%I A346119 #7 Jul 05 2021 15:04:50
%S A346119 1,0,1,2,0,-16,-35,342,2779,-6424,-239382,-822460,22393657,278844084,
%T A346119 -1553468891,-68399947042,-275025888900,15302175612416,
%U A346119 243541868882077,-2463105309082902,-121649966081262521,-473088821582805820,50905612811064360006,945133249101683013812,-15321255878414345388335
%N A346119 Expansion of the e.g.f. sqrt(2*x*exp(x) - 2*exp(x) + 3).
%F A346119 E.g.f. y(x) satisfies y*y' = x*exp(x).
%F A346119 a(0)=1, a(n) = Sum_{k=1..floor(n/2)} (-1)^(k-1)*A006677(k)*A008306(n,k) for n > 0.
%F A346119 For all p prime, a(p) == -1 (mod p).
%F A346119 For n > 1, a(n) == 0 (mod (n-1)).
%F A346119 Conjecture: a(n) = 0 for only n = 1 and n = 4.
%e A346119 sqrt(2*x*exp(x)-2*exp(x)+3) = 1 + x^2/2! + 2*x^3/3! - 16*x^5/5! - 35*x^6/6! + 342*x^7/7! + 2779*x^8/8! - 6424*x^9/9! + ...
%e A346119 a(11) = Sum_{k=1..5} (-1)^(k-1)*A006677(k)*A008306(11,k) = -822460.
%e A346119 For k=1, (-1)^(1-1)*A006677(1)*A008306(11,1) == -1 (mod 11), because A006677(1) = 1 and A008306(11,1) = (11-1)!
%e A346119 For k>=2, (-1)^(k-1)*A006677(k)*A008306(11,k) == 0 (mod 11), because A008306(11,k) == 0 (mod 11), result a(11) == -1 (mod 11).
%e A346119 a(8) = Sum_{k=1..4} (-1)^(k-1)*A006677(k)*A008306(8,k) = 2779.
%e A346119 a(8) == 0 (mod (8-1)), because for k >= 1, A008306(8,k) == 0 (mod 7).
%p A346119 stirtr:= proc(p) proc(n) add(p(k)*Stirling2(n, k), k=0..n) end end: f:= n-> `if`(n=0, 1, (2*n-2)!/ (n-1)!/ 2^(n-1)): A006677:= stirtr(f): # Alois P. Heinz, 2008.
%p A346119 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 A346119 a:= n-> add(((-1)^(k-1)*A006677(k)*A008306(n,k)), k=1..iquo(n,2)):a(0):=1 ; seq(a(n), n=0..24);
%p A346119 # second program:
%p A346119 a := series(sqrt(2*x*exp(x)-2*exp(x)+3), x=0, 25):seq(n!*coeff(a, x, n), n=0..24);
%t A346119 CoefficientList[Series[Sqrt(2*x*E^x-2*E^x+3), {x, 0, 24}], x] * Range[0, 24]!
%o A346119 (PARI) my(x='x+O('x^30)); Vec(serlaplace(sqrt(2*x*exp(x) - 2*exp(x) + 3))) \\ _Michel Marcus_, Jul 05 2021
%Y A346119 Cf. A000166, A006677, A008306, A327006, A345652, A345697, A345969.
%K A346119 sign
%O A346119 0,4
%A A346119 _Mélika Tebni_, Jul 05 2021