OFFSET
0,2
LINKS
Manuel Kauers, Algorithms for D-finite functions.
Wikipedia, Holonomic_function
FORMULA
a(n) = sqrt(2/Pi)*Sum_{k=0..n} s(k)*s(n-k-1/2) where s(n) = 2^n*(Gamma(n/2 + 1/2)/Gamma(n/2 + 1))^cos(n*Pi).
a(n) = [x^n] (4*x^2 - x - 1)/((1 - 4*x^2)^(3/2)*(2*x - 1)).
a(n) ~ 2^n * n^(3/2) / (3*sqrt(2*Pi)). - Vaclav Kotesovec, Oct 19 2019
D-finite with recurrence: n*a(n) +(n-4)*a(n-1) +2*(-4*n+3)*a(n-2) +4*(-n+2)*a(n-3) +16*(n-2)*a(n-4)=0. - R. J. Mathar, Jan 09 2020
D-finite with recurrence: a(n) = (4*n*(n*(n + 5) - 10)*a(n-2) + 2*(n - 6)*(n + 2)*a(n-1))/(n*(n*(n + 3) - 14)). - Peter Luschny, Jan 10 2020
Let q(n) = a(n+1)/a(n) and d(n) = numerator(q(n)) - 2*denominator(q(n)).
Conjecture: d(n) is periodic with period 12 and repetition pattern (1, 2, 3, 3, 3, 2, 1, 1, 3, 6, 3, 1). - Peter Luschny, Jan 10 2020
a(n) = Sum_{k=0..n} 2^(n-k)*A056040(k). - Peter Luschny, Apr 22 2021
MAPLE
s := z -> 2^z*(GAMMA(z/2+1/2)/GAMMA(z/2+1))^cos(z*Pi):
a := n -> sqrt(2/Pi)*add(s(k)*s(n-k-1/2), k=0..n): seq(a(n), n=0..19);
# Alternative:
a := proc(n) option remember; if n=0 then 1 elif n=1 then 3 else
(4*n*(n*(n+5)-10)*a(n-2) + 2*(n-6)*(n+2)*a(n-1))/(n*(n*(n+3)-14)) fi end:
seq(a(n), n=0..31); # Peter Luschny, Jan 10 2020
CROSSREFS
KEYWORD
nonn
AUTHOR
Peter Luschny, Oct 02 2019
EXTENSIONS
Simpler name by Peter Luschny, Apr 22 2021
STATUS
approved