A338632 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.

1, 1, 2, 14, 166, 2714, 55866, 1377942, 39493518, 1288115570, 47086272754, 1906554619166, 84711219819062, 4098314765667082, 214489189682087594, 12075596389435432230, 727783484288200558110, 46755528594469120151010, 3189788089674119448202722

Offset

0,3

Links

Paul D. Hanna, Table of n, a(n) for n = 0..200

Formula

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

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

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).

a(n) ~ 2^(2*n) * n^(n - 1/2) / (sqrt(Pi) * exp(n + 1/2)). - Vaclav Kotesovec, Nov 12 2020

Example

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 + ...
where
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.

Prog

(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] }
for(n=0, 20, print1(a(n), ", "))

Crossrefs

Cf. A000699, A158119, A338636.

Cf. A191107, A186776.

Sequence in context: A277362 A209937 A245896 * A124215 A355781 A003582

Adjacent sequences: A338629 A338630 A338631 * A338633 A338634 A338635

Keyword

nonn

Author

Paul D. Hanna, Nov 04 2020

Status

approved