A338632 - OEIS

%I #13 Nov 12 2020 11:12:56

%S 1,1,2,14,166,2714,55866,1377942,39493518,1288115570,47086272754,

%T 1906554619166,84711219819062,4098314765667082,214489189682087594,

%U 12075596389435432230,727783484288200558110,46755528594469120151010,3189788089674119448202722

%N G.f. A(x) satisfies: 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - ...))))), a continued fraction relation.

%H Paul D. Hanna, <a href="/A338632/b338632.txt">Table of n, a(n) for n = 0..200</a>

%F a(n) = 2 (mod 4) for n > 1 (conjecture).

%F For n > 0, a(n) = 1 (mod 3) iff n = A191107(k) for some k >= 1 (conjecture).

%F For n > 0, a(n) = 2 (mod 3) iff n = A186776(k) for some k >= 2 where A186776 is the Stanley sequence S(0,2) (conjecture).

%F a(n) ~ 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * exp(n + 1/2)). - _Vaclav Kotesovec_, Nov 12 2020

%e G.f. A(x) = 1 + x + 2*x^2 + 14*x^3 + 166*x^4 + 2714*x^5 + 55866*x^6 + 1377942*x^7 + 39493518*x^8 + 1288115570*x^9 + 47086272754*x^10 + ...

%e where

%e 1 = A(x) - x/(A(x) - 3*x/(A(x) - 5*x/(A(x) - 7*x/(A(x) - 9*x/(A(x) - 11*x/(A(x) - 13*x/(A(x) - 15*x/(A(x) - 17*x/(A(x) - 19*x/(A(x) - ...)))))))))), a continued fraction relation.

%o (PARI) {a(n) = my(A=[1],CF=1); for(i=1,n, A=concat(A,0); for(i=1,#A, CF = Ser(A) - (2*(#A-i)+1)*x/CF ); A[#A] = -polcoeff(CF,#A-1) );A[n+1] }

%o for(n=0,20,print1(a(n),", "))

%Y Cf. A000699, A158119, A338636.

%Y Cf. A191107, A186776.

%K nonn

%O 0,3

%A _Paul D. Hanna_, Nov 04 2020