OFFSET
0,3
FORMULA
a(0) = 0; a(n) = (2*n-1)*a(n-1) + (-1)^(n-1).
From Peter Bala, Feb 10 2024: (Start)
a(n) = (2*n - 2)*a(n-1) + (2*n - 3)*a(n-2) with a(0) = 0 and a(1) = 1.
The double factorial numbers (2*n-1)!! = A001147(n) satisfy the same recurrence, leading to the generalized continued fraction expansion Limit_{n -> oo} a(n)/(2*n-1)!! = Sum_{k >= 1} (-1)^(k-1)/(2*k-1)!! = 0.7247784590... = 1 - 1/(3 + 3/(4 + 5/(6 + 7/(8 + 9/(10 + ... )))))). (End)
PROG
(PARI) a001147(n) = prod(k=1, n, 2*k-1);
a(n) = a001147(n)*sum(k=1, n, (-1)^(k-1)/a001147(k));
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Seiichi Manyama, Jan 06 2024
STATUS
approved