A061572 a(n) = (n!)^2 * Sum_{k=1..n} 1/(k^2*(k-1)!).

1, 5, 47, 758, 18974, 683184, 33476736, 2142516144, 173543847984, 17354385161280, 2099880608143680, 302382807612606720, 51102694487009537280, 10016128119460096327680, 2253628826878608852019200, 576928979680925173791283200, 166732475127787396148470732800

Offset: 1

Author

N. J. A. Sloane, May 19 2001

Keywords

• nonn

Links

• Harry J. Smith, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A061573.

Programs

PARI

a(n) = { n!^2*sum(k=1, n, 1/(k^2*(k-1)!)) } \\ Harry J. Smith, Jul 24 2009

Formula

Recurrence: a(1) = 1, a(2) = 5, a(n) = (n^2+n-1)*a(n-1) - (n-1)^3*a(n-2) for n >= 3. The sequence b(n) = n!^2 also satisfies this recurrence with the initial conditions b(1) = 1 and b(2) = 4. Hence we have the finite continued fraction expansion a(n)/b(n) = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)))). Lim n -> infinity a(n)/b(n) = Ei(1) - gamma = 1/(1-1^3/(5-2^3/(11-...-(n-1)^3/(n^2+n-1)-...))). Cf. A061573. - Peter Bala, Jul 10 2008