A368794 a(n) = (2*n-1)!! * Sum_{k=1..n} (-1)^(k-1)/(2*k-1)!!.

0, 1, 2, 11, 76, 685, 7534, 97943, 1469144, 24975449, 474533530, 9965204131, 229199695012, 5729992375301, 154709794133126, 4486584029860655, 139084104925680304, 4589775462547450033, 160642141189160751154, 5943759223998947792699, 231806609735958963915260

Offset

0,3

Links

Table of n, a(n) for n=0..20.

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

Cf. A001147, A286286, A306858.

Sequence in context: A350680 A349408 A053481 * A110329 A221844 A006766

Adjacent sequences: A368791 A368792 A368793 * A368795 A368796 A368797

Keyword

nonn,easy

Author

Seiichi Manyama, Jan 06 2024

Status

approved