OFFSET
1,2
FORMULA
a(n) = (n - 1)! * ((1 - sqrt(3))^n + (1 + sqrt(3))^n) / 2^n.
D-finite with recurrence +2*a(n) +2*(-n+1)*a(n-1) -(n-1)*(n-2)*a(n-2)=0. - R. J. Mathar, Aug 20 2021
MAPLE
b:= proc(n) b(n):= n! * (<<1|1>, <1/2|0>>^n)[1, 1] end:
a:= proc(n) option remember; `if`(n=0, 0, b(n)-add(
binomial(n, j)*j*b(n-j)*a(j), j=1..n-1)/n)
end:
seq(a(n), n=1..25); # Alois P. Heinz, Oct 11 2019
MATHEMATICA
nmax = 21; CoefficientList[Series[-Log[1 - x - x^2/2], {x, 0, nmax}], x] Range[0, nmax]! // Rest
FullSimplify[Table[(n - 1)! ((1 - Sqrt[3])^n + (1 + Sqrt[3])^n)/2^n, {n, 1, 21}]]
PROG
(PARI) my(x='x+O('x^25)); Vec(serlaplace(-log(1 - x - x^2/2))) \\ Michel Marcus, Oct 11 2019
CROSSREFS
KEYWORD
nonn
AUTHOR
Ilya Gutkovskiy, Oct 11 2019
STATUS
approved