OFFSET
0,3
COMMENTS
a(n) == 0 (mod 7^k) for n >= 7*k, for k>=1 (conjecture).
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..390
FORMULA
E.g.f.: ( (1 + x/S) / (1 - S*x) )^(sqrt(2)/4) where S = sqrt(2) + 1.
E.g.f.: exp( Integral 1/(1-2*x-x^2) dx ).
a(n) ~ n! * 2^(3*sqrt(2)/8) * n^(sqrt(2)/4-1) * (1+sqrt(2))^(n-1/(2*sqrt(2))) / GAMMA(1/(2*sqrt(2))). - Vaclav Kotesovec, Jun 28 2014
a(0) = a(1) = 1; a(n) = (2*n-1) * a(n-1) + (n-1) * (n-2) * a(n-2). - Ilya Gutkovskiy, Aug 13 2021
E.g.f.: exp((1/sqrt(2)) * arctanh(x*sqrt(2)/(1-x))). - Fabian Pereyra, Oct 11 2023
a(n) = n!*Sum_{k=0..n} binomial(n-1,k-1)*binomial(1/sqrt(8),k)*(1+sqrt(2))^(n-k)*(sqrt(8))^k. - Fabian Pereyra, Oct 19 2023
EXAMPLE
E.g.f.: A(x) = 1 + x + 3*x^2 + 17*x^3/3! + 137*x^4/4! + 1437*x^5/5! + ...
where
log(A(x)) = x + 2*x^2/2 + 5*x^3/3 + 12*x^4/4 + 29*x^5/5 + 70*x^6/6 + 169*x^7/7 + 408*x^8/8 + 985*x^9/9 + ... + A000129(n)*x^n/n + ...
PROG
(PARI) {a(n)=n!*polcoeff(exp(intformal(1/(1-2*x-x^2 +x*O(x^n)))), n)}
for(n=0, 30, print1(a(n), ", "))
CROSSREFS
KEYWORD
nonn
AUTHOR
Paul D. Hanna, Jun 27 2014
STATUS
approved