OFFSET
0,3
FORMULA
a(n) = [x^n] (1/(2*x))*(1 - (4*x - 1)/(sqrt((6*x - 1)*(2*x - 1)))).
D-finite with recurrence a(n) = 4*(3*(n^2 - n)*a(n - 2) - (2*n^2 - n)*a(n - 1))/(1 - n^2) for n >= 2.
The INVERT transform of A052177.
a(n) ~ 2^(n - 1/2) * 3^(n + 1/2) / sqrt(Pi*n). - Vaclav Kotesovec, May 12 2021
MAPLE
gf := exp(2*x)*BesselI(1, x):
ser := series(gf, x, 32): seq(2^n*n!*coeff(ser, x, n), n = 0..23);
# Or:
gf := (1/(2*x))*(1 - (4*x - 1)/(sqrt((6*x - 1)*(2*x - 1)))):
ser := series(gf, x, 32): seq(coeff(ser, x, n), n = 0..23);
MATHEMATICA
RecurrenceTable[{(1 - n^2) a[n] == 4 (3 (n^2 - n) a[n - 2] - (2 n^2 - n) a[n - 1]), a[0] == 0, a[1] == 1}, a, {n, 0, 23}]
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Peter Luschny, May 12 2021
STATUS
approved