%I #13 Oct 28 2024 05:52:48
%S 1,2,9,62,577,6762,95569,1581398,29985249,640943954,15247684249,
%T 399543448878,11434012567201,354818583385658,11867268458655393,
%U 425562496103250758,16288135278501487681,662740598471482775202,28565932640460309847081,1300233394242252193354526
%N E.g.f.: exp( Sum_{n>=1} Pell(n+1)*x^n/n ), where Pell(n) = A000129(n).
%F E.g.f.: A(x) = 1/(1-2*x-x^2) * exp( Sum_{n>=1} -Pell(n-1)*x^n/n ).
%F E.g.f.: exp((1/sqrt(2))*arctanh(x*sqrt(2)/(1-x)))/sqrt(1-2*x-x^2). - _Fabian Pereyra_, Oct 19 2024
%F a(n) ~ 2^(3/2^(5/2) - 1/4) * sqrt(Pi) * (1 + sqrt(2))^(n + 1/2 - 1/2^(3/2)) * n^(n + 1/2^(3/2)) / (Gamma(1/2 + 1/2^(3/2)) * exp(n)). - _Vaclav Kotesovec_, Oct 28 2024
%e E.g.f.: A(x) = 1 + 2*x + 9*x^2/2! + 62*x^3/3! + 577*x^4/4! + 6762*x^5/5! +...
%e such that
%e log(A(x)) = 2*x + 5*x^2/2 + 12*x^3/3 + 29*x^4/4 + 70*x^5/5 + 169*x^6/6 + 408*x^7/7 + 985*x^8/8 +...+ A000129(n+1)*x^n/n +...
%t nmax = 20; CoefficientList[Series[((1 - x + Sqrt[2]*x)/(1 - x - Sqrt[2]*x))^2^(-3/2) / Sqrt[1 - 2*x - x^2], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Oct 28 2024 *)
%o (PARI) {Pell(n)=polcoeff(x/(1-2*x-x^2 +x*O(x^n)),n)}
%o {a(n)=n!*polcoeff(exp(sum(m=1,n,Pell(m+1)*x^m/m)+x*O(x^n)),n)}
%o for(n=0,30,print1(a(n),", "))
%Y Cf. A000129.
%K nonn
%O 0,2
%A _Paul D. Hanna_, Jun 13 2012